# Line Integrals (Exercises)

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**1.*** **True or False?* Line integral \(\displaystyle\int _C f(x,y)\,ds\) is equal to a definite integral if \(C\) is a smooth curve defined on \([a,b]\) and if function \(f\) is continuous on some region that contains curve \(C\).

**Answer**- True

**2.*** True or False?* Vector functions \(\vecs r_1=t\,\hat{\mathbf i}+t^2\,\hat{\mathbf j}, \quad 0≤t≤1,\) and \(\vecs r_2=(1−t)\,\hat{\mathbf i}+(1−t)^2\,\hat{\mathbf j}, \quad 0≤t≤1\), define the same oriented curve.

**3. ***True or False?* \(\displaystyle\int _{−C}(P\,dx+Q\,dy)=\int _C(P\,dx−Q\,dy)\)

**Answer**- False

**4. ***True or False?* A piecewise smooth curve \(C\) consists of a finite number of smooth curves that are joined together end to end.

**5. ***True or False?* If \(C\) is given by \(x(t)=t,\quad y(t)=t, \quad 0≤t≤1\), then \(\displaystyle\int _Cxy\,ds=\int ^1_0t^2\,dt.\)

**Answer**- False

**For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path.**

**6.** **[T]** \(\displaystyle\int _C(x+y)\,ds\)

\(C:x=t,y=(1−t),z=0\) from \((0, 1, 0)\) to \((1, 0, 0)\)

**7. [T]** \(\displaystyle \int _C(x−y)ds\)

\(C:\vecs r(t)=4t\,\hat{\mathbf i}+3t\,\hat{\mathbf j}\) when \(0≤t≤2\)

**Answer**- \(\displaystyle\int _C(x−y)\,ds=10\)

**8. [T]** \(\displaystyle\int _C(x^2+y^2+z^2)\,ds\)

\(C:\vecs r(t)=\sin t\,\hat{\mathbf i}+\cos t\,\hat{\mathbf j}+8t\,\hat{\mathbf k}\) when \(0≤t≤\dfrac{π}{2}\)

**9. [T]** Evaluate \(\displaystyle\int _Cxy^4\,ds\), where \(C\) is the right half of circle \(x^2+y^2=16\) and is traversed in the clockwise direction.

**Answer**- \(\displaystyle\int _Cxy^4\,ds=\frac{8192}{5}\)

**10. [T]** Evaluate \(\displaystyle\int _C4x^3ds\), where *C* is the line segment from \((−2,−1)\) to \((1, 2)\).

**For the following exercises, find the work done.**

**11. **Find the work done by vector field \(\vecs F(x,y,z)=x\,\hat{\mathbf i}+3xy\,\hat{\mathbf j}−(x+z)\,\hat{\mathbf k}\) on a particle moving along a line segment that goes from \((1,4,2)\) to \((0,5,1)\).

**Answer**- \(W=8\) units of work

**12.** Find the work done by a person weighing 150 lb walking exactly one revolution up a circular, spiral staircase of radius 3 ft if the person rises 10 ft.

**13. **Find the work done by force field \(\vecs F(x,y,z)=−\dfrac{1}{2}x\,\hat{\mathbf i}−\dfrac{1}{2}y\,\hat{\mathbf j}+\dfrac{1}{4}\,\hat{\mathbf k}\) on a particle as it moves along the helix \(\vecs r(t)=\cos t\,\hat{\mathbf i}+\sin t\,\hat{\mathbf j}+t\,\hat{\mathbf k}\) from point \((1,0,0)\) to point \((−1,0,3π)\).

**Answer**- \(W=\frac{3π}{4}\) units of work

**14.** Find the work done by vector field \(\vecs{F}(x,y)=y\,\hat{\mathbf i}+2x\,\hat{\mathbf j}\) in moving an object along path \(C\), which joins points \((1, 0)\) and \((0, 1)\).

**15. **Find the work done by force \(\vecs{F}(x,y)=2y\,\hat{\mathbf i}+3x\,\hat{\mathbf j}+(x+y)\,\hat{\mathbf k}\) in moving an object along curve \(\vecs r(t)=\cos(t)\,\hat{\mathbf i}+\sin(t)\,\hat{\mathbf j}+16\,\hat{\mathbf k}\), where \(0≤t≤2π\).

**Answer**- \(W=π\) units of work

**16. **Find the mass of a wire in the shape of a circle of radius 2 centered at (3, 4) with linear mass density \(ρ(x,y)=y^2\).

**For the following exercises, evaluate the line integrals.**

**17.** Evaluate \(\displaystyle\int_C\vecs F·d\vecs{r}\), where \(\vecs{F}(x,y)=−1\,\hat{\mathbf j}\), and \(C\) is the part of the graph of \(y=\frac{1}{2}x^3−x\) from \((2,2)\) to \((−2,−2)\).

**Answer**- \(\displaystyle\int _C\vecs F·d\vecs{r}=4\) units of work

**18.** Evaluate \(\displaystyle\int _γ(x^2+y^2+z^2)^{−1}ds\), where \(γ\) is the helix \(x=\cos t,y=\sin t,z=t,\) with \(0≤t≤T.\)

**19.** Evaluate \(\displaystyle\int _Cyz\,dx+xz\,dy+xy\,dz\) over the line segment from \((1,1,1) \) to \((3,2,0).\)

**Answer**- \(\displaystyle\int _Cyz\,dx+xz\,dy+xy\,dz=−1\)

**20. **Let \(C\) be the line segment from point (0, 1, 1) to point (2, 2, 3). Evaluate line integral \(\displaystyle\int _Cy\,ds.\)

**21. [T]** Use a computer algebra system to evaluate the line integral \(\displaystyle\int _Cy^2\,dx+x\,dy\), where \(C\) is the arc of the parabola \(x=4−y^2\) from \((−5, −3)\) to \((0, 2)\).

**Answer**- \(\displaystyle\int _C(y^2)\,dx+(x)\,dy=\dfrac{245}{6}\)

**22. [T]** Use a computer algebra system to evaluate the line integral \(\displaystyle\int _C (x+3y^2)\,dy\) over the path \(C\) given by \(x=2t,y=10t,\) where \(0≤t≤1.\)

**23. [T]** Use a CAS to evaluate line integral \(\displaystyle\int _C xy\,dx+y\,dy\) over path \(C\) given by \(x=2t,y=10t\), where \(0≤t≤1\).

**Answer**- \(\displaystyle\int _Cxy\,dx+y\,dy=\dfrac{190}{3}\)

**24.** Evaluate line integral \(\displaystyle\int _C(2x−y)\,dx+(x+3y)\,dy\), where \(C\) lies along the \(x\)-axis from \(x=0\) to \(x=5\).

**26. [T]** Use a CAS to evaluate \(\displaystyle\int _C\dfrac{y}{2x^2−y^2}\,ds\), where \(C\) is defined by the parametric equations \(x=t,y=t\), for \(1≤t≤5.\)

**Answer**- \(\displaystyle\int _C\frac{y}{2x^2−y^2}\,ds=\sqrt{2}\ln 5\)

**27. [T]** Use a CAS to evaluate \(\displaystyle\int _Cxy\,ds\), where \(C\) is defined by the parametric equations \(x=t^2,y=4t\), for \(0≤t≤1.\)

**In the following exercises, find the work done by force field \(\vecs F\) on an object moving along the indicated path.**

**28. **\(\vecs{F}(x,y)=−x \,\hat{\mathbf i}−2y\,\hat{\mathbf j}\)

\(C:y=x^3\) from \((0, 0)\) to \((2, 8)\)

**Answer**- \(W=−66\) units of work

**29. **\(\vecs{F}(x,y)=2x\,\hat{\mathbf i}+y\,\hat{\mathbf j}\)

<\(C\): counterclockwise around the triangle with vertices \((0, 0), (1, 0), \) and \((1, 1)\)

**30.** \(\vecs F(x,y,z)=x\,\hat{\mathbf i}+y\,\hat{\mathbf j}−5z\,\hat{\mathbf k}\)

\(C:\vecs r(t)=2\cos t\,\hat{\mathbf i}+2\sin t\,\hat{\mathbf j}+t\,\hat{\mathbf k},\; 0≤t≤2π\)

**Answer**- \(W=−10π^2\) units of work

**31.** Let \(\vecs F\) be vector field \(\vecs{F}(x,y)=(y^2+2xe^y+1)\,\hat{\mathbf i}+(2xy+x^2e^y+2y)\,\hat{\mathbf j}\). Compute the work of integral \(\displaystyle\int _C\vecs F·d\vecs{r}\), where \(C\) is the path \(\vecs r(t)=\sin t\,\hat{\mathbf i}+\cos t\,\hat{\mathbf j},\quad 0≤t≤\dfrac{π}{2}\).

**32.** Compute the work done by force \(\vecs F(x,y,z)=2x\,\hat{\mathbf i}+3y\,\hat{\mathbf j}−z\,\hat{\mathbf k}\) along path \(\vecs r(t)=t\,\hat{\mathbf i}+t^2\,\hat{\mathbf j}+t^3\,\hat{\mathbf k}\), where \(0≤t≤1\).

**Answer**- \(W=2\) units of work

**33. **Evaluate \(\displaystyle\int _C\vecs F·d\vecs{r}\), where \(\vecs{F}(x,y)=\dfrac{1}{x+y}\,\hat{\mathbf i}+\dfrac{1}{x+y}\,\hat{\mathbf j}\) and \(C\) is the segment of the unit circle going counterclockwise from \((1,0)\) to \((0, 1)\).

**34. **Force \(\vecs F(x,y,z)=zy\,\hat{\mathbf i}+x\,\hat{\mathbf j}+z^2x\,\hat{\mathbf k}\) acts on a particle that travels from the origin to point \((1, 2, 3)\). Calculate the work done if the particle travels:

- along the path \((0,0,0)→(1,0,0)→(1,2,0)→(1,2,3)\) along straight-line segments joining each pair of endpoints;
- along the straight line joining the initial and final points.
- Is the work the same along the two paths?

**Answer**- a. \(W=11\) units of work;

b. \(W=\dfrac{39}{4}=9\frac{3}{4}\) units of work;

c. No

**35.** Find the work done by vector field \(\vecs F(x,y,z)=x\,\hat{\mathbf i}+3xy\,\hat{\mathbf j}−(x+z)\,\hat{\mathbf k}\) on a particle moving along a line segment that goes from \((1, 4, 2)\) to \((0, 5, 1).\)

**36.** How much work is required to move an object in vector field \(\vecs{F}(x,y)=y\,\hat{\mathbf i}+3x\,\hat{\mathbf j}\) along the upper part of ellipse \(\dfrac{x^2}{4}+y^2=1\) from \((2, 0)\) to \((−2,0)\)?

**Answer**- \(W=2π\) units of work

**37. **A vector field is given by \(\vecs{F}(x,y)=(2x+3y)\,\hat{\mathbf i}+(3x+2y)\,\hat{\mathbf j}\). Evaluate the line integral of the field around a circle of unit radius traversed in a clockwise fashion.

**38. **Evaluate the line integral of scalar function \(xy\) along parabolic path \(y=x^2\) connecting the origin to point \((1, 1)\).

**Answer**- \(\displaystyle\int _C f\,ds=\dfrac{25\sqrt{5}+1}{120}\)

**39.** Find \(\displaystyle\int _Cy^2\,dx+(xy−x^2)\,dy\) along \(C: y=3x\) from \((0, 0)\) to \((1, 3).\)

**40.** Find \(\displaystyle\int _Cy^2\,dx+(xy−x^2)\,dy\) along \(C: y^2=9x\) from \((0, 0)\) to \((1, 3).\)

**Answer**- \(\displaystyle\int _Cy^2\,dx+(xy−x^2)\,dy=6.15\)

**For the following exercises, use a CAS to evaluate the given line integrals.**

**41. [T]** Evaluate \(\vecs F(x,y,z)=x^2z\,\hat{\mathbf i}+6y\,\hat{\mathbf j}+yz^2\,\hat{\mathbf k}\), where \(C\) is represented by \(\vecs r(t)=t\,\hat{\mathbf i}+t^2\,\hat{\mathbf j}+\ln t \,\hat{\mathbf k},1≤t≤3\).

**42. [T]** Evaluate line integral \(\displaystyle\int _γxe^y\,ds\) where, \(γ\) is the arc of curve \(x=e^y\) from \((1,0)\) to \((e,1)\).

**Answer**- \(\displaystyle\int _γxe^y\,ds≈7.157\)

**43. [T]** Evaluate the integral \(\displaystyle\int _γxy^2\,ds\), where \(γ\) is a triangle with vertices \((0, 1, 2), (1, 0, 3)\), and \((0,−1,0)\).

**44. [T]** Evaluate line integral \(\displaystyle\int _γ(y^2−xy)\,dx\), where \(γ\) is curve \(y=\ln x\) from \((1, 0)\) toward \((e,1)\).

**Answer**- \(\displaystyle\int _γ(y^2−xy)\,dx≈−1.379\)

**45. [T]** Evaluate line integral \(\displaystyle\int_γ xy^4\,ds\), where \(γ\) is the right half of circle \(x^2+y^2=16\).

**46. [T]** Evaluate \(\int C \vecs F⋅d\vecs{r},\int C \vecs F·d\vecs{r},\) where \(\vecs F(x,y,z)=x^2y\,\mathbf{\hat i}+(x−z)\,\mathbf{\hat j}+xyz\,\mathbf{\hat k}\) and

\(C: \vecs r(t)=t\,\mathbf{\hat i}+t^2\,\mathbf{\hat j}+2\,\mathbf{\hat k},0≤t≤1\).

**Answer**- \(\displaystyle\int _C \vecs F⋅d\vecs{r}≈−1.133\) units of work

**47.** Evaluate \(\displaystyle\int _C \vecs F⋅d\vecs{r}\), where \(\vecs{F}(x,y)=2x\sin y\,\mathbf{\hat i}+(x^2\cos y−3y^2)\,\mathbf{\hat j}\) and

\(C\) is any path from \((−1,0)\) to \((5, 1)\).

**48. **Find the line integral of \(\vecs F(x,y,z)=12x^2\,\mathbf{\hat i}−5xy\,\mathbf{\hat j}+xz\,\mathbf{\hat k}\) over path \(C\) defined by \(y=x^2, z=x^3\) from point \((0, 0, 0)\) to point \((2, 4, 8)\).

**Answer**- \(\displaystyle\int _C \vecs F⋅d\vecs{r}≈22.857\) units of work

**49.** Find the line integral of \(\displaystyle\int _C(1+x^2y)\,ds\), where \(C\) is ellipse \(\vecs r(t)=2\cos t\,\mathbf{\hat i}+3\sin t\,\mathbf{\hat j}\) from \(0≤t≤π.\)

**For the following exercises, find the flux.**

**50.** Compute the flux of \(\vecs{F}=x^2\,\mathbf{\hat i}+y\,\mathbf{\hat j}\) across a line segment from \((0, 0)\) to \((1, 2).\)

**Answer**- \(\text{flux}=−\frac{1}{3}\)

**51.** Let \(\vecs{F}=5\,\mathbf{\hat i}\) and let \(C\) be curve \(y=0,\) with \(0≤x≤4\). Find the flux across \(C\).

**52.** Let \(\vecs{F}=5\,\mathbf{\hat j}\) and let \(C\) be curve \(y=0,\) with \(0≤x≤4\). Find the flux across \(C\).

**Answer**- \(\text{flux}=-20\)

**53.** Let \(\vecs{F}=−y\,\mathbf{\hat i}+x\,\mathbf{\hat j}\) and let \(C: \vecs r(t)=\cos t\,\mathbf{\hat i}+\sin t\,\mathbf{\hat j}\) for \(0≤t≤2π\). Calculate the flux across \(C\).

**54. **Let \(\vecs{F}=(x^2+y^3)\,\mathbf{\hat i}+(2xy)\,\mathbf{\hat j}\). Calculate flux \(\vecs F\) orientated counterclockwise across the curve \(C: x^2+y^2=9.\)

**Answer**- \(\text{flux}=0\)

**Complete the rest of the exercises as stated.**

**55.** Find the line integral of \(\displaystyle\int _C z^2\,dx+y\,dy+2y\,dz,\) where \(C\) consists of two parts: \(C_1\) and \(C_2.\) \(C_1\) is the intersection of cylinder \(x^2+y^2=16\) and plane \(z=3\) from \((0, 4, 3)\) to \((−4,0,3).\) \(C_2\) is a line segment from \((−4,0,3)\) to \((0, 1, 5)\).

**56.** A spring is made of a thin wire twisted into the shape of a circular helix \(x=2\cos t,\;y=2\sin t,\;z=t.\) Find the mass of two turns of the spring if the wire has a constant mass density of \(ρ\) grams per cm.

**Answer**- \(m=4πρ\sqrt{5}\) grams

**57.** A thin wire is bent into the shape of a semicircle of radius \(a\). If the linear mass density at point \(P\) is directly proportional to its distance from the line through the endpoints, find the mass of the wire.

**58.** An object moves in force field \(\vecs F(x,y,z)=y^2\,\mathbf{\hat i}+2(x+1)y\,\mathbf{\hat j}\) counterclockwise from point \((2, 0)\) along elliptical path \(x^2+4y^2=4\) to \((−2,0)\), and back to point \((2, 0)\) along the \(x\)-axis. How much work is done by the force field on the object?

**Answer**- \(W=0\) units of work

**59. **Find the work done when an object moves in force field \(\vecs F(x,y,z)=2x\,\mathbf{\hat i}−(x+z)\,\mathbf{\hat j}+(y−x)\,\mathbf{\hat k}\) along the path given by \(\vecs r(t)=t^2\,\mathbf{\hat i}+(t^2−t)\,\mathbf{\hat j}+3\,\mathbf{\hat k}, \; 0≤t≤1.\)

**60. **If an inverse force field \(\vecs F\) is given by \(\vecs F(x,y,z)=\dfrac{k}{‖r‖^3}r\), where \(k\) is a constant, find the work done by \(\vecs F\) as its point of application moves along the \(x\)-axis from \(A(1,0,0)\) to \(B(2,0,0)\).

**Answer**- \(W=\frac{k}{2}\) units of work

**61.** David and Sandra plan to evaluate line integral \(\displaystyle\int _C\vecs F·d\vecs{r}\) along a path in the \(xy\)-plane from \((0, 0)\) to \((1, 1)\). The force field is \(\vecs{F}(x,y)=(x+2y)\,\mathbf{\hat i}+(−x+y^2)\,\mathbf{\hat j}\). David chooses the path that runs along the \(x\)-axis from \((0, 0)\) to \((1, 0)\) and then runs along the vertical line \(x=1\) from \((1, 0)\) to the final point \((1, 1)\). Sandra chooses the direct path along the diagonal line \(y=x\) from \((0, 0)\) to \((1, 1)\). Whose line integral is larger and by how much?

## Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.