9.7E: Exercises
- Page ID
- 25946
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exercise \(\PageIndex{1}\): Transformation
In the following exercises, the function \(T : S \rightarrow R, \space T (u,v) = (x,y)\) on the region \(S = \{(u,v) | 0 \leq u \leq 1, \space 0 \leq v \leq 1\}\) bounded by the unit square is given, where \(R \in R^2\) is the image of \(S\) under \(T\).
a. Justify that the function \(T\) is a \(C^1\) transformation.
b. Find the images of the vertices of the unit square \(S\) through the function \(T\).
c. Determine the image \(R\) of the unit square \(S\) and graph it.
1. \(x = 2u, \space y = 3v\)
2. \(x = \frac{u}{2}, \space y = \frac{v}{3}\)
3. \(x = u - v, \space y = u + v\)
4. \(x = 2u - v, \space y = u + 2v\)
5. \(x = u^2, \space y = v^2\)
6. \(x = u^3, \space y = v^3\)
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2.
a. \(T(u,v) = (g(u,v), \space h(u,v), \space x = g(u,v) = \frac{u}{2}\) and \(y = h(u,v) = \frac{v}{3}\). The functions \(g\) and \(h\) are continuous and differentiable, and the partial derivatives \(g_u (u,v) = \frac{1}{2}, \space g_v (u,v) = 0, \space h_u (u,v) = 0\) and \(h_v (u,v) = \frac{1}{3}\) are continuous on \(S\);
b. \(T(0,0) = (0,0), \space T(1,0) = \left(\frac{1}{2},0\right), \space T(0,1) = \left(0,\frac{1}{3}\right)\), and \(T(1,1) = \left(\frac{1}{2}, \frac{1}{3} \right)\);
c. \(R\) is the rectangle of vertices \((0,0), \space \left(0,\frac{1}{3}\right), \space \left(\frac{1}{2}, \frac{1}{3} \right)\), and \(\left(0,\frac{1}{3}\right)\) in the \(xy\)-plane; the following figure.
4.
a. \(T(u,v) = (g(u,v), \space h(u,v), \space x = g(u,v) = 2u - v\) and \(y = h(u,v) = u + 2v\). The functions \(g\) and \(h\) are continuous and differentiable, and the partial derivatives \(g_u (u,v) = 2, \space g_v (u,v) = -1, \space h_u (u,v) = 1\) and \(h_v (u,v) = 2\) are continuous on \(S\);
b. \(T(0,0) = (0,0), \space T(1,0) = (2,1), \space T(0,1) = (-1,2)\), and \(T(1,1) = (1,3)\);
c. \(R\) is the parallelogram of vertices \((0,0), \space (2,1) \space (1,3)\), and \((-1,2)\) in the \(xy\)-plane; the following figure.
6.
a. \(T(u,v) = (g(u,v), \space h(u,v), \space x = g(u,v) = u^3\) and \(y = h(u,v) = v^3\). The functions \(g\) and \(h\) are continuous and differentiable, and the partial derivatives \(g_u (u,v) = 3u^2, \space g_v (u,v) = 0, \space h_u (u,v) = 0\) and \(h_v (u,v) = 3v^2\) are continuous on \(S\);
b. \(T(0,0) = (0,0), \space T(1,0) = (1,0), \space T(0,1) = (0,1)\), and \(T(1,1) = (1,1)\);
c. \(R\) is the unit square in the \(xy\)-plane see the figure in the answer to the previous exercise.
Exercise \(\PageIndex{2}\): One to one
In the following exercises, determine whether the transformations \(T : S \rightarrow R\) are one-to-one or not.
1. \(x = u^2, \space y = v^2\), where \(S\) is the rectangle of vertices \((-1,0), \space (1,0), \space (1,1)\), and \((-1,1)\).
2. \(x = u^4, \space y = u^2 + v\), where \(S\) is the triangle of vertices \((-2,0), \space (2,0)\), and \((0,2)\).
3. \(x = 2u, \space y = 3v\), where \(S\) is the square of vertices \((-1,1), \space (-1,1), \space (-1,-1)\), and \((1,-1)\).
4. \(x = u + v + w, \space y = u + v, \space z = w\), where \(S = R = R^3\).
5. \(x = u^2 + v + w, \space y = u^2 + v, \space z = w\), where \(S = R = R^3\).
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2. \(T\) is not one-to-one: two points of \(S\) have the same image. Indeed, \(T(-2,0) = T(2,0) = (16,4)\).
3. \(T\) is one-to-one: We argue by contradiction. \(T(u_1,v_1) = T(u_2,v_2)\) implies \(2u_1 - v_1 = 2u_2 - v_2\) and \(u_1 = u_2\). Thus, \(u_1 = u+2\) and \(v_1 = v_2\).
5. \(T\) is not one-to-one: \(T(1,v,w) = (-1,v,w)\)
Exercise \(\PageIndex{3}\): Inverse Transformation
In the following exercises, the transformations \(T : R \rightarrow S\) are one-to-one. Find their related inverse transformations \(T^{-1} : R \rightarrow S\).
- \(x = 4u, \space y = 5v\), where \(S = R = R^2\).
- \(x = u + 2v, \space y = -u + v\), where \(S = R = R^2\).
- \(x = e^{2u+v}, \space y = e^{u-v}\), where \(S = R^2\) and \(R = \{(x,y) | x > 0, \space y > 0\}\)
- \(x = \ln u, \space y = \ln(uv)\), where \(S = \{(u,v) | u > 0, \space v > 0\}\) and \(R = R^2\).
- \(x = u + v + w, \space y = 3v, \space z = 2w\), where \(S = R = R^3\).
- \(x = u + v, \space y = v + w, \space z = u + w\), where \(S = R = R^3\).
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2. \(u = \frac{x-2y}{3}, \space v= \frac{x+y}{3}\).
4. \(u = e^x, \space v = e^{-x+y}\).
6. \(u = \frac{x-y+z}{2}, \space v = \frac{x+y-z}{2}, \space w = \frac{-x+y+z}{2}\).
Exercise \(\PageIndex{4}\)
In the following exercises, the transformation \(T : S \rightarrow R, \space T (u,v) = (x,y)\) and the region \(R \subset R^2\) are given. Find the region \(S \subset R^2\).
- \(x = au, \space y = bv, \space R = \{(x,y) | x^2 + y^2 \leq a^2 b^2\}\) where \(a,b > 0\)
- \(x = au, \space y = bc, \space R = \{(x,y) | \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1\}\), where \(a,b > 0\)
- \(S = \{(u,v) | u^2 + v^2 \leq 1\}\)
- \(x = \frac{u}{a}, \space y = \frac{v}{b}, \space z = \frac{w}{c}, \space R = \{(x,y)|x^2 + y^2 + z^2 \leq 1\}\), where \(a,b,c > 0\)
- \(x = au, \space y = bv, \space z = cw, \space R = \{(x,y)|\frac{x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2} \leq 1, \space z > 0\}\), where \(a,b,c > 0\)
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2.\(S = \{(u,v) | u^2 + v^2 \leq 1\}\).
5. \(R = \{(u,v,w)|u^2 - v^2 - w^2 \leq 1, \space w > 0\}\)
Exercise \(\PageIndex{5}\): Jacobian
In the following exercises, find the Jacobian \(J\) of the transformation.
- \(x = u + 2v, \space y = -u + v\)
- \(x = \frac{u^3}{2}, \space y = \frac{v}{u^2}\)
- \(x = e^{2u-v}, \space y = e^{u+v}\)
- \(x = ue^v, \space y = e^{-v}\)
- \(x = u \space \cos (e^v), \space y = u \space \sin(e^v)\)
- \(x = v \space \sin (u^2), \space y = v \space \cos(u^2)\)
- \(x = u \space \cosh v, \space y = u \space \sinh v, \space z = w\)
- \(x = v \space \cosh \left(\frac{1}{u}\right), \space y = v \space \sinh \left(\frac{1}{u}\right), \space z = u + w^2\)
- \(x = u + v, \space y = v + w, \space z = u\)
- \(x = u - v, \space y = u + v, \space z = u + v + w\)
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2.\(\frac{3}{2}\)
4. \(-1\)
6. \(2uv\)
8. \(\frac{v}{u^2}\)
10. \(2\)
Exercise \(\PageIndex{6}\)
1. The triangular region \(R\) with the vertices \((0,0), \space (1,1)\), and \((1,2)\) is shown in the following figure.
a. Find a transformation \(T : S \rightarrow R, \space T(u,v) = (x,y) = (au + bv + dv)\), where \(a,b,c\), and \(d\) are real numbers with \(ad - bc \neq 0\) such that \(T^{-1} (0,0) = (0,0), \space T^{-1} (1,1) = (1,0)\), and \(T^{-1}(1,2) = (0,1)\).
b. Use the transformation \(T\) to find the area \(A(R)\) of the region \(R\).
2.
The triangular region \(R\) with the vertices \((0,0), \space (2,0)\), and \((1,3)\) is shown in the following figure.
a. Find a transformation \(T : S \rightarrow R, \space T(u,v) = (x,y) = (au + bv + dv)\), where \(a,b,c\), and \(d\) are real numbers with \(ad - bc \neq 0\) such that \(T^{-1} (0,0) = (0,0), \space T^{-1} (2,0) = (1,0)\), and \(T^{-1}(1,3) = (0,1)\).
b. Use the transformation \(T\) to find the area \(A(R)\) of the region \(R\).
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2.
a. \(T (u,v) = (2u + v, \space 3v); b. The area of \(R\) is
\[A(R) = \int_0^3 \int_{y/3}^{(6-y)/3} dx \space dy = \int_0^1 \int_0^{1-u} \left|\frac{\partial (x,y)}{\partial (u,v)}\right| dv \space du = \int_0^1 \int_0^{1-u} 6 dv \space du = 3.\]
Exercise \(\PageIndex{7}\)
In the following exercises, use the transformation \(u = y - x, \space v = y\), to evaluate the integrals on the parallelogram \(R\) of vertices \((0,0), \space (1,0), \space (2,1)\), and \((1,1)\) shown in the following figure.
1. \[\iint_R (y - x) dA\]
2. \[\iint_R (y^2 - xy)dA\]
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\(-\frac{1}{4}\)
Exercise \(\PageIndex{8}\)
In the following exercises, use the transformation \(y = x = u, \space x + y = v\) to evaluate the integrals on the square \(R\) determined by the lines \(y = x, \space y = -x + 2, \space y = x + 2\), and \(y = -x\) shown in the following figure.
\[\iint_R e^{x+y} dA\]
\[\iint_R \sin (x - y) dA\]
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\(-1 + cos 2\)
Exercise \(\PageIndex{9}\)
In the following exercises, use the transformation \(x = u, \space 5y = v\) to evaluate the integrals on the region \(R\) bounded by the ellipse \(x^2 + 25y^2 = 1\) shown inthe following figure.
\[\iint_R \sqrt{x^2 + 25y^2} dA\]
\[\iint_R (x^2 + 25y^2)^2 dA\]
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\(\frac{\pi}{15}\)
Exercise \(\PageIndex{10}\)
In the following exercises, use the transformation \(u = x + y, \space v = x - y\) to evaluate the integrals on the trapezoidal region \(R\) determined by the points \((1,0), \space (2,0), \space (0,2)\), and \((0,1)\) shown in the following figure.
\[\iint_R (x^2 - 2xy + y^2) \space e^{x+y} dA\]
\[\iint_R (x^3 + 3x^2y + 3xy^2 + y^3) \space dA\]
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\(\frac{31}{5}\)
Exercise \(\PageIndex{11}\)
1. The circular annulus sector \(R\) bounded by the circles \(4x^2 + 4y^2 = 1\) and \(9x^2 + 9y^2 = 64\), the line \(x = y \sqrt{3}\), and the \(y\)-axis is shown in the following figure. Find a transformation \(T\) from a rectangular region \(S\) in the \(r\theta\)-plane to the region \(R\) in the \(xy\)-plane. Graph \(S\).
2. The solid \(R\) bounded by the circular cylinder \(x^2 + y^2 = 9\) and the planes \(z = 0, \space z = 1, \space x = 0\), and \(y = 0\) is shown in the following figure. Find a transformation \(T\) from a cylindrical box \(S\) in \(r\theta z\)-space to the solid \(R\) in \(xyz\)-space.
3. Show that \[\iint_R f \left(\sqrt{\frac{x^2}{3} + \frac{y^2}{3}}\right) dA = 2 \pi \sqrt{15} \int_0^1 f (\rho) \rho \space d\rho,\] where \(f\) is a continuous function on \([0,1]\) and \(R\) is the region bounded by the ellipse \(5x^2 + 3y^2 = 15\).
4. Show that \[\iiint_R f \left(\sqrt{16x^2 + 4y^2 + z^2}\right) dV = \frac{\pi}{2} \int_0^1 f (\rho) \rho^2 d\rho,\] where \(f\) is a continuous function on \([0,1]\) and \(R\) is the region bounded by the ellipsoid \(16x^2 + 4y^2 + z^2 = 1\).
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2. \(T (r,\theta,z) = (r \space \cos \theta, \space r \space \sin \theta, \space z); \space S = [0,3] \times [0,\frac{\pi}{2}] \times [0,1]\) in the \(r\theta z\)-space
Exercise \(\PageIndex{12}\)
1. [T] Find the area of the region bounded by the curves \(xy = 1, \space xy = 3, \space y = 2x\), and \(y = 3x\) by using the transformation \(u = xy\) and \(v = \frac{y}{x}\). Use a computer algebra system (CAS) to graph the boundary curves of the region \(R\).
2. [T] Find the area of the region bounded by the curves \(x^2y = 2, \space x^2y = 3, \space y = x\), and \(y = 2x\) by using the transformation \(u = x^2y\) and \(v = \frac{y}{x}\). Use a CAS to graph the boundary curves of the region \(R\).
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2. The area of \(R\) is \(10 - 4\sqrt{6}\); the boundary curves of \(R\) are graphed in the following figure.
Exercise \(\PageIndex{13}\)
1. Evaluate the triple integral \[\int_0^1 \int_1^2 \int_z^{z+1} (y + 1) \space dx \space dy \space dz\] by using the transformation \(u = x - z, \space v = 3y\), and \(w = \frac{z}{2}\).
2. Evaluate the triple integral \[\int_0^2 \int_4^6 \int_{3z}^{3z+2} (5 - 4y) \space dx \space dy \space dz\] by using the transformation \(u = x - 3z, \space v = 4y\), and \(w = z\).
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2. \(8\)
Exercise \(\PageIndex{14}\)
1. A transformation \(T : R^2 \rightarrow R^2, \space T (u,v) = (x,y)\) of the form \(x = au + bv, \space y = cu + dv\), where \(a,b,c\), and \(d\) are real numbers, is called linear. Show that a linear transformation for which \(ad - bc \neq 0\) maps parallelograms to parallelograms.
2. A transformation \(T_{\theta} : R^2 \rightarrow R^2, \space T_{\theta} (u,v) = (x,y)\) of the form \(x = u \space \cos \theta - v \space \sin \theta, \space y = u \space \sin \theta + v \space \cos \theta\), is called a rotation angle \(\theta\). Show that the inverse transformation of \(T_{\theta}\) satisfies \(T_{\theta}^{-1} = T_{-\theta}\) where \(T_{-\theta}\) is the rotation of angle \(-\theta\).
Exercise \(\PageIndex{15}\)
1. [T] Find the region \(S\) in the \(uv\)-plane whose image through a rotation of angle \(\frac{\pi}{4}\) is the region \(R\) enclosed by the ellipse \(x^2 + 4y^2 = 1\). Use a CAS to answer the following questions.
a. Graph the region \(S\).
b. Evaluate the integral \[\iint_S e^{-2uv} du \space dv.\] Round your answer to two decimal places.
2. [T] The transformations \(T_i : \mathbb{R}^2 \rightarrow \mathbb{R}^2, \space i = 1, . . . , 4,\) defined by \(T_1(u,v) = (u,-v), \space T_2 (u,v) = (-u,v), \space T_3 (u,v) = (-u, -v)\), and \(T_4 (u,v) = (v,u)\) are called reflections about the \(x\)-axis, \(y\)-axis origin, and the line \(y = x\), respectively.
a. Find the image of the region \(S = \{(u,v)|u^2 + v^2 - 2u - 4v + 1 \leq 0\}\) in the \(xy\)-plane through the transformation \(T_1 \circ T_2 \circ T_3 \circ T_4\).
b. Use a CAS to graph \(R\).
c. Evaluate the integral \[\iint_S \sin (u^2) \space du \space dv\] by using a CAS. Round your answer to two decimal places.
3. [T] The transformations \(T_{k,1,1} : \mathbb{R}^3 \rightarrow \mathbb{R}^3, \space T_{k,1,1}(u,v,w) = (x,y,z)\) of the form \(x = ku, \space y = v, \space z = w\), where \(k \neq 1\) is a positive real number, is called a stretch if \(k > 1\) and a compression if \(0 < k < 1\) in the \(x\)-direction. Use a CAS to evaluate the integral \[\iiint_S e^{-(4x^2+9y^2+25z^2)} dx \space dy \space dz\] on the solid \(S = \{(x,y,z) | 4x^2 + 9y^2 + 25z^2 \leq 1\}\) by considering the compression \(T_{2,3,5}(u,v,w) = (x,y,z)\) defined by \(x = \frac{u}{2}, \space y = \frac{v}{3}\), and \(z = \frac{w}{5}\). Round your answer to four decimal places.
4. [T] The transformation \(T_{a,0} : \mathbb{R}^2 \rightarrow \mathbb{R}^2, \space T_{a,0} (u,v) = (u + av, v)\), where \(a \neq 0\) is a real number, is called a shear in the \(x\)-direction. The transformation, \(T_{b,0} : R^2 \rightarrow R^2, \space T_{o,b}(u,v) = (u,bu + v)\), where \(b \neq 0\) is a real number, is called a shear in the \(y\)-direction.
a. Find transformations \(T_{0,2} \circ T_{3,0}\).
b. Find the image \(R\) of the trapezoidal region \(S\) bounded by \(u = 0, \space v = 0, \space v = 1\), and \(v = 2 - u\) through the transformation \(T_{0,2} \circ T_{3,0}\).
c. Use a CAS to graph the image \(R\) in the \(xy\)-plane.
d. Find the area of the region \(R\) by using the area of region \(S\).
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2. a. \(R = \{(x,y)|y^2 + x^2 - 2y - 4x + 1 \leq 0\}\); b. \(R\) is graphed in the following figure;
c. \(3.16\)
4.
a. \(T_{0,2} \circ T_{3,0}(u,v) = (u + 3v, 2u + 7v)\);
b. The image \(S\) is the quadrilateral of vertices \((0,0), \space (3,7), \space (2,4)\), and \((4,9)\);
c. \(S\) is graphed in the following figure;
d. \(\frac{3}{2}\)
Exercise \(\PageIndex{16}\)
1. Use the transformation, \(x = au, \space y = av, \space z = cw\) and spherical coordinates to show that the volume of a region bounded by the spheroid \(\frac{x^2+y^2}{a^2} + \frac{z^2}{c^2} = 1\) is \(\frac{4\pi a^2c}{3}\).
2. Find the volume of a football whose shape is a spheroid \(\frac{x^2+y^2}{a^2} + \frac{z^2}{c^2} = 1\) whose length from tip to tip is \(11\) inches and circumference at the center is \(22\) inches. Round your answer to two decimal places.
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2. \(\frac{2662}{3\pi} \approx 282.45 \space in^3\)