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1.1E: Exercises

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Determining Volumes by Slicing

Exercise $$\PageIndex{1}$$

Derive the formula for the volume of a sphere using the slicing method.

As in most problems we should choose to graph the function first. We know the formula for a circle, $$x^2+y^2=r^2$$ and solving for y we get $$y=\pm\sqrt{R^2-x^2}$$. When graphed they appear as: With the blue line being the positive values and the green the negatives. Now we must find what an area of a slice will be. As we are trying to find the volume of a sphere the area of a slice must be a circle, with radius $$y=r=\sqrt{R^2-x^2}$$.

Setting up the area of a slice we get $A(x)=\pi r^2=\pi (\sqrt{R^2-x^2})^2=\pi(R^2-x^2)$

Integrating over the interval of $$[-R,R]$$ we get: $V=\pi \int_{-R}^{R}(R^2-x^2)dx=\pi \Big[R^2x-\dfrac{1}{3}x^3\Big]_{-R}^{R}=\dfrac{4}{3}\pi R^3$

Exercise $$\PageIndex{2}$$

Derive the formula for the volume of a cone.

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{3}$$

Use the slicing method to derive the formula for the volume of a tetrahedron with side length a.

TBD.

Exercise $$\PageIndex{4}$$

Use the disk method to derive the formula for the volume of a trapezoidal cylinder.

TBD

Exercise $$\PageIndex{5}$$

Explain when you would use the disk method versus the washer method. When are they interchangeable?

TBD

For exercises 6 - 10, draw a typical slice and find the volume using the slicing method for the given volume.

Exercise $$\PageIndex{6}$$

A pyramid with height 6 units and square base of side 2 units, as pictured here. Here the cross-sections are squares taken perpendicular to the $$y$$-axis.
We use the vertical cross-section of the pyramid through its center to obtain an equation relating $$x$$ and $$y$$.
Here this would be the equation, $$y = 6 - 6x$$. Since we need the dimensions of the square at each $$y$$-level, we solve this equation for $$x$$ to get, $$x = 1 - \tfrac{y}{6}$$.
This is half the distance across the square cross-section at the $$y$$-level, so the side length of the square cross-section is, $$s = 2\left(1 - \tfrac{y}{6}\right).$$
Thus, we have the area of a cross-section is,

$$A(y) = \left[2\left(1 - \tfrac{y}{6}\right)\right]^2 = 4\left(1 - \tfrac{y}{6}\right)^2.$$

\begin{align*} \text{Then},\quad V &= \int_0^6 4\left(1 - \tfrac{y}{6}\right)^2 \, dy \\[5pt] &= -24 \int_1^0 u^2 \, du, \quad \text{where} \, u = 1 - \tfrac{y}{6}, \, \text{so} \, du = -\tfrac{1}{6}\,dy, \quad \implies \quad -6\,du = dy \\[5pt] &= 24 \int_0^1 u^2 \, du = 24\dfrac{u^3}{3}\bigg|_0^1 \\[5pt] &= 8u^3\bigg|_0^1 \\[5pt] &= 8\left( 1^3 - 0^3 \right) \quad= \quad 8\, \text{units}^3 \end{align*}

Exercise $$\PageIndex{7}$$

A pyramid with height 4 units and a rectangular base with length 2 units and width 3 units, as pictured here. TBD

Exercise $$\PageIndex{8}$$

A tetrahedron with a base side of 4 units,as seen here. $$V = \frac{32}{3\sqrt{2}} = \frac{16\sqrt{2}}{3}$$ units3

Exercise $$\PageIndex{9}$$

A pyramid with height 5 units, and an isosceles triangular base with lengths of 6 units and 8 units, as seen here. TBD

Exercise $$\PageIndex{10}$$

A cone of radius $$r$$ and height $$h$$ has a smaller cone of radius $$r/2$$ and height $$h/2$$ removed from the top, as seen here. The resulting solid is called a frustum. $$V = \frac{7\pi}{12} hr^2$$ units3

For exercises 11 - 16, draw an outline of the solid and find the volume using the slicing method.

Exercise $$\PageIndex{11}$$

The base is a circle of radius $$a$$. The slices perpendicular to the base are squares.

TBD

Exercise $$\PageIndex{12}$$

The base is a triangle with vertices $$(0,0),(1,0),$$ and $$(0,1)$$. Slices perpendicular to the $$xy$$-plane are semicircles. $$\displaystyle V = \int_0^1 \frac{\pi(1-x)^2}{8}\, dx \quad = \quad \frac{π}{24}$$ units3

Exercise $$\PageIndex{13}$$

The base is the region under the parabola $$y=1−x^2$$ in the first quadrant. Slices perpendicular to the $$xy$$-plane are squares.

TBD

Exercise $$\PageIndex{14}$$

The base is the region under the parabola $$y=1−x^2$$ and above the $$x$$-axis. Slices perpendicular to the $$y$$-axis are squares. $$\displaystyle V = \int_0^1 4(1 - y)\,dy \quad = \quad 2$$ units3

Exercise $$\PageIndex{15}$$

The base is the region enclosed by $$y=x^2)$$ and $$y=9.$$ Slices perpendicular to the $$x$$-axis are right isosceles triangles.

TBD

Exercise $$\PageIndex{16}$$

The base is the area between $$y=x$$ and $$y=x^2$$. Slices perpendicular to the $$x$$-axis are semicircles. $$\displaystyle V = \int_0^1 \frac{\pi}{8}\left( x - x^2 \right)^2 \, dx \quad=\quad \frac{π}{240}$$ units3

Disk and Washer Method

For exercises 17 - 24, draw the region bounded by the curves. Then, use the disk or washer method to find the volume when the region is rotated around the $$x$$-axis.

Exercise $$\PageIndex{17}$$

$$x+y=8,\quad x=0$$, and $$y=0$$

TBD

Exercise $$\PageIndex{18}$$

$$y=2x^2,\quad x=0,\quad x=4,$$ and $$y=0$$ $$\displaystyle V = \int_0^4 4\pi x^4\, dx \quad=\quad \frac{4096π}{5}$$ units3

Exercise $$\PageIndex{19}$$

$$y=e^x+1,\quad x=0,\quad x=1,$$ and $$y=0$$

TBD

Exercise $$\PageIndex{20}$$

$$y=x^4,\quad x=0$$, and $$y=1$$ $$\displaystyle V = \int_0^1 \pi\left( 1^2 - \left( x^4\right)^2\right)\, dx = \int_0^1 \pi\left( 1 - x^8\right)\, dx \quad = \quad \frac{8π}{9}$$ units3

Exercise $$\PageIndex{21}$$

$$y=\sqrt{x},\quad x=0,\quad x=4,$$ and $$y=0$$

TBD

Exercise $$\PageIndex{22}$$

$$y=\sin x,\quad y=\cos x,$$ and $$x=0$$ $$\displaystyle V = \int_0^{\pi/4} \pi \left( \cos^2 x - \sin^2 x\right) \, dx = \int_0^{\pi/4} \pi \cos 2x \, dx \quad=\quad \frac{π}{2}$$ units3

Exercise $$\PageIndex{23}$$

$$y=\dfrac{1}{x},\quad x=2$$, and $$y=3$$

TBD

Exercise $$\PageIndex{24}$$

$$x^2−y^2=9$$ and $$x+y=9,\quad y=0$$ and $$x=0$$ $$V = 207π$$ units3

For exercises 25 - 32, draw the region bounded by the curves. Then, find the volume when the region is rotated around the $$y$$-axis.

Exercise $$\PageIndex{25}$$

$$y=4−\dfrac{1}{2}x,\quad x=0,$$ and $$y=0$$

TBD

Exercise $$\PageIndex{26}$$

$$y=2x^3,\quad x=0,\quad x=1,$$ and $$y=0$$ $$V = \frac{4π}{5}$$ units3

Exercise $$\PageIndex{27}$$

$$y=3x^2,\quad x=0,$$ and $$y=3$$

TBD

Exercise $$\PageIndex{28}$$

$$y=\sqrt{4−x^2},\quad y=0,$$ and $$x=0$$ $$V = \frac{16π}{3}$$ units3

Exercise $$\PageIndex{29}$$

$$y=\dfrac{1}{\sqrt{x+1}},\quad x=0$$, and $$x=3$$

TBD

Exercise $$\PageIndex{30}$$

$$x=\sec(y)$$ and $$y=\dfrac{π}{4},\quad y=0$$ and $$x=0$$ $$V = π$$ units3

Exercise $$\PageIndex{31}$$

$$y=\dfrac{1}{x+1},\quad x=0$$, and $$x=2$$

TBD

Exercise $$\PageIndex{32}$$

$$y=4−x,\quad y=x,$$ and $$x=0$$ $$V = \frac{16π}{3}$$ units3

For exercises 33 - 40, draw the region bounded by the curves. Then, find the volume when the region is rotated around the $$x$$-axis.

Exercise $$\PageIndex{33}$$

$$y=x+2,\quad y=x+6,\quad x=0$$, and $$x=5$$

TBD

Exercise $$\PageIndex{34}$$

$$y=x^2$$ and $$y=x+2$$ $$V = \frac{72π}{5}$$ units3

Exercise $$\PageIndex{35}$$

$$x^2=y^3$$ and $$x^3=y^2$$

TBD

Exercise $$\PageIndex{36}$$

$$y=4−x^2$$ and $$y=2−x$$ $$V = \frac{108π}{5}$$ units3

Exercise $$\PageIndex{37}$$

$$y=\cos x,\quad y=e^{−x},\quad x=0$$, and $$x=1.2927$$

TBD

Exercise $$\PageIndex{38}$$

$$y=\sqrt{x}$$ and $$y=x^2$$ $$V = \frac{3π}{10}$$ units3

Exercise $$\PageIndex{39}$$

$$y=\sin x,\quad y=5\sin x,\quad x=0$$ and $$x=π$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{40}$$

$$y=\sqrt{1+x^2}$$ and $$y=\sqrt{4−x^2}$$ $$V = 2\sqrt{6}π$$ units3

For exercises 41 - 45, draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the $$y$$-axis.

Exercise $$\PageIndex{41}$$

$$y=\sqrt{x},\quad x=4$$, and $$y=0$$

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Exercise $$\PageIndex{42}$$

$$y=x+2,\quad y=2x−1$$, and $$x=0$$ $$V = 9π$$ units3

Exercise $$\PageIndex{43}$$

$$y=\dfrac{3}{x}$$ and $$y=x^3$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{44}$$

$$x=e^{2y},\quad x=y^2,\quad y=0$$, and $$y=\ln(2)$$ $$V = \dfrac{π}{20}(75−4\ln^5(2))$$ units3

Exercise $$\PageIndex{45}$$

$$x=\sqrt{9−y^2},\quad x=e^{−y},\quad y=0$$, and $$y=3$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{46}$$

Yogurt containers can be shaped like frustums. Rotate the line $$y=\left(\frac{1}{m}\right)x$$ around the $$y$$-axis to find the volume between $$y=a$$ and $$y=b$$. $$V = \dfrac{m^2π}{3}(b^3−a^3)$$ units3

Exercise $$\PageIndex{47}$$

Rotate the ellipse $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ around the $$x$$-axis to approximate the volume of a football, as seen here. Add texts here. Do not delete this text first.

Exercise $$\PageIndex{48}$$

Rotate the ellipse $$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$$ around the $$y$$-axis to approximate the volume of a football.

$$V = \frac{4a^2bπ}{3}$$ units3

Exercise $$\PageIndex{49}$$

A better approximation of the volume of a football is given by the solid that comes from rotating $$y=\sin x) around the \(x$$-axis from $$x=0$$ to $$x=π$$. What is the volume of this football approximation, as seen here? Add texts here. Do not delete this text first.

Exercise $$\PageIndex{50}$$

What is the volume of the Bundt cake that comes from rotating $$y=\sin x$$ around the $$y$$-axis from $$x=0$$ to $$x=π$$? $$V = 2π^2$$ units3

For exercises 51 - 56, find the volume of the solid described.

Exercise $$\PageIndex{51}$$

The base is the region between $$y=x$$ and $$y=x^2$$. Slices perpendicular to the $$x$$-axis are semicircles.

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{52}$$

The base is the region enclosed by the generic ellipse $$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1.$$ Slices perpendicular to the $$x$$-axis are semicircles.

$$V = \frac{2ab^2π}{3}$$ units3

Exercise $$\PageIndex{53}$$

Bore a hole of radius a down the axis of $$a$$ right cone and through the base of radius $$b$$, as seen here. Add texts here. Do not delete this text first.

Exercise $$\PageIndex{54}$$

Find the volume common to two spheres of radius $$r$$ with centers that are $$2h$$ apart, as shown here. $$V = \frac{π}{12}(r+h)^2(6r−h)$$ units3

Exercise $$\PageIndex{55}$$

Find the volume of a spherical cap of height $$h$$ and radius $$r$$ where $$h<r$$, as seen here. Add texts here. Do not delete this text first.

Exercise $$\PageIndex{56}$$

Find the volume of a sphere of radius $$R$$ with a cap of height $$h$$ removed from the top, as seen here. $$V = \dfrac{π}{3}(h+R)(h−2R)^2$$ units3