Double Integrals

Exercises

1. Compute the value of each of the following double integrals on the indicated rectangular region \(R.\) For practice, be sure you also sketch/imagine the expanse (solid) of which each integral computes the volume.

   \(\displaystyle \iint_R x-3y^2 \,\mathrm{d}A \\ R = \{(x,y) \mid 0\leq x\leq 2, 1\leq y\leq 2\}\)

   \(\displaystyle \iint_R y\sin(xy) \,\mathrm{d}A \\ R = [1,2]\!\times\![0,\pi]\)

   \(\displaystyle \iint_R 7x^3y \,\mathrm{d}A \\ R = [1,3]\!\times\![-1,2]\)

   \(\displaystyle \iint_R \sqrt{1-x^2} \,\mathrm{d}A \\ R = [-1,1]\!\times\![-2,2]\)

   \(\displaystyle \iint_R y\mathrm{e}^{xy} \,\mathrm{d}A \\ R = [1,2]\!\times\![1,10]\)

   \(\displaystyle \iint_R \frac{1}{(x+2y)^2} \,\mathrm{d}A \\ R = [1,2]\!\times\![2,5]\)

   \(\displaystyle \iint_R \frac{y^2+1}{x^2+1} \,\mathrm{d}A \\ R = [0,1]\!\times\![0,1]\)

   \(\displaystyle \iint_R \frac{x}{y}+\frac{y}{x} \,\mathrm{d}A \\ R = [1,3]\!\times\![1,5]\)

   \(\displaystyle \iint_R \frac{\ln(y)}{xy} \,\mathrm{d}A \\ R = [1,5]\!\times\![1,3]\)

   \(\displaystyle \iint_R \sqrt{x+y} \,\mathrm{d}A \\ R = [0,1]\!\times\![0,1]\)

   \(\displaystyle \iint_R xy\sqrt{x^2+y^2} \,\mathrm{d}A \\ R = [0,1]\!\times\![0,1]\)

   \(\displaystyle \iint_R \mathrm{e}^{3x+5y} \,\mathrm{d}A \\ R = [0,2]\!\times\![0,2]\)

   \(\displaystyle \iint_R \frac{xy^2}{1+x^2} \,\mathrm{d}A \\ R = [0,1]\!\times\![-2,2]\)

   \(\displaystyle \iint_R y\cos(x+y) \,\mathrm{d}A \\ R = [0,\tfrac{\pi}{4}]\!\times\![0, \tfrac{\pi}{3}]\)

2. For each of the following iterated integrals, sketch/imagine the expanse (solid) of which volume the integral computes.

   \( \displaystyle \int\limits_{-1}^{1}\int\limits_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} \sqrt{1-x^2-y^2} \,\mathrm{d}y\,\mathrm{d}x\)

   \( \displaystyle \int\limits_{-1}^{1}\int\limits_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} 1-x^2-y^2 \,\mathrm{d}y\,\mathrm{d}x\)

   \( \displaystyle \int\limits_{-1}^{1}\int\limits_{0}^{\sqrt{1-x^2}} \sqrt{1-x^2-y^2} \,\mathrm{d}y\,\mathrm{d}x\)

   \( \displaystyle \int\limits_{-1}^{1}\int\limits_{-1}^{1} \sqrt{1-x^2} \,\mathrm{d}x\,\mathrm{d}y\)

   \( \displaystyle \int\limits_{-1}^{1}\int\limits_{-1}^{1} 3+x+2y \,\mathrm{d}x\,\mathrm{d}y\)

   \( \displaystyle \int\limits_{0}^{1}\int\limits_{0}^{1-y} 1-x-y \,\mathrm{d}x\,\mathrm{d}y\)

   \( \displaystyle \int\limits_{-2}^{2}\int\limits_{-1}^{3-x^2} \mathrm{e}^{-y} \,\mathrm{d}y\,\mathrm{d}x\)

3. Compute the volume of the solid \(S\) that is bounded by the elliptic paraboloid \({x^2+2y^2+z=16}\) the planes \(x=2\) and \(y=2,\) and the coordinate planes.
4. Evaluate \(\iint_R x+2y \,\mathrm{d}A\) where \(R\) is the region in the \(xy\)-plane bounded by the parabolas \(y=2x^2\) and \(y=1+x^2.\)
5. Evaluate \(\iint_R x^2+xy \,\mathrm{d}A\) where \(R\) is the triangular region in the \(xy\)-plane with vertices located at \((0,0)\) and \((1,1)\) and \((3,0).\)
6. Evaluate \(\iint_R x+y \,\mathrm{d}A\) where \(R\) is the region in the \(xy\)-plane bound between the graphs of the \(y=\sqrt{x}\) and \(y=x-8.\)
7. The graphs of \(\sin(x)\) and \(\cos(x)\) bound infinitely many congruent copies of a single shape. What is the area of this shape?
8. Evaluate \(\iint_R xy \,\mathrm{d}A\) where \(R\) is the region in the \(xy\)-plane bounded by the parabolas \(y = 2x^2-3x+4\) and \(y=x^2+x+1.\)
9. Evaluate \(\iint_R 3x+2y \,\mathrm{d}A\) where \(R\) consists of the two regions in the \(xy\)-plane bounded by the graph of \(f(x) = x^3-6x^2+11x-6\) and the \(x\)-axis.
10. Evaluate \(\iint_R \ln(xy) \,\mathrm{d}A\) where \(R\) is the triangular region in the \(xy\)-plane with vertices located at \((1,1)\) and \((2,5)\) and \((3,4).\)
11. Evaluate \(\iint_R \frac{x}{1+y^2} \,\mathrm{d}A\) where \(R\) is the region in the \(xy\)-plane bounded by the curves \(x = \sqrt{y}\) and \(x=0\) and \(y=9.\)
12. Evaluate \(\iint_R x\sqrt{(x+y)(x-y)} \,\mathrm{d}A\) where \(R\) is the region in the \(xy\)-plane bounded by the lines \(y=x\) and \(x=0\) and \(y=2.\)
13. Evaluate \(\iint_R \mathrm{e}^{xy} \,\mathrm{d}A\) where \(R\) is the region in the \(xy\)-plane bounded by the curves \(y = \frac{1}{x}\) and \(x=1\) and \(y=\mathrm{e}.\)
14. Evaluate \(\iint_R \frac{1}{(xy)^2+1} \,\mathrm{d}A\) where \(R\) is the region in the \(xy\)-plane bounded by the curves \(xy = \ln(x)\) and \(xy=1\) and \(x=1.\)
15. Evaluate \(\iint_R \ln(xy) \,\mathrm{d}A\) where \(R\) is the region in the \(xy\)-plane bounded by the curves \(xy=1\) and \(x+2y=3.\)
16. Evaluate \(\iint_R y\tan(x) \,\mathrm{d}A\) where \(R\) is the region in the \(xy\)-plane bounded by the graph of the secant function between \(\frac{-\pi}{2}\) and \(\frac{\pi}{2}\) and the line \(y=2.\)
17. Evaluate \(\iint_R y\cos^3(x) \,\mathrm{d}A\) where \(R\) is the region in the \(xy\)-plane bounded by the \(x\)-axis and the graph of \(\sin^2(x)\) between \(x=0\) and \(x=\pi.\)
18. Evaluate \(\iint_R \frac{x-1}{y(x+2)} \,\mathrm{d}A\) where \(R\) is the region in the \(xy\)-plane bounded by the lines \(x=y-1\) and \(x=3\) and \(y=2.\)

19. Evaluate each of the following iterated integrals. Hint: each integral is difficult (or impossible) to evaluate as written. Instead, evaluate the equivalent integral with the order of integration swapped.

    \( \displaystyle \int\limits_0^1\int\limits_x^1 \sin\big(y^2\big) \,\mathrm{d}y\,\mathrm{d}x\)

    \(\displaystyle \int\limits_{0}^{1}\int\limits_{\sqrt{y}}^{1} \frac{1}{\sqrt{x^3+1}} \,\mathrm{d}x\,\mathrm{d}y\)

    \(\displaystyle \int\limits_{0}^{1}\int\limits_{x}^{1} \mathrm{e}^{x/y} \,\mathrm{d}y\,\mathrm{d}x\)

    \(\displaystyle \int\limits_{0}^{8}\int\limits_{\sqrt[3]{y}}^{2} \mathrm{e}^{x^4} \,\mathrm{d}x\,\mathrm{d}y\)

    \(\displaystyle \int\limits_{1}^{\mathrm{e}}\int\limits_{\ln(y)}^{1} \cos\bigl(\mathrm{e}^x-x\bigr) \,\mathrm{d}x\,\mathrm{d}y\)

    \(\displaystyle \int\limits_{0}^{6}\int\limits_{x/3}^{2} x\sqrt{y^3+1} \,\mathrm{d}y\,\mathrm{d}x\)

20. Rewrite each of the following integrals in polar coordinates before evaluating.

    \( \displaystyle \int\limits_{-1}^{1}\int\limits_{0}^{\sqrt{1-x^2}} x^2+y^2 \,\mathrm{d}y\,\mathrm{d}x\)

    \(\displaystyle \int\limits_{-5}^{5}\int\limits_{0}^{\sqrt{25-x^2}} \cos\bigl(x^2+y^2\bigr) \,\mathrm{d}y\,\mathrm{d}x\)

    \(\displaystyle \int\limits_{0}^{2}\int\limits_{0}^{\sqrt{2x-x^2}} \sqrt{x^2+y^2} \,\mathrm{d}y\,\mathrm{d}x\)

    \(\displaystyle \int\limits_{0}^{7}\int\limits_{-\sqrt{49-y^2}}^{0} x^2y \,\mathrm{d}x\,\mathrm{d}y\)

21. Compute the volume of each of the following solids. Hint: sketch or imagine the solid first and decide whether an integral in rectangular coordinates or polar coordinates will be simpler to evaluate.

    1. The solid that lies under the paraboloid \(z = x^2+y^2\) and above the region \(R\) in the \(xy\)-plane bounded by \(y=2x\) and \(y=x^2.\)
    2. The tetrahedron bounded by the planes \(x+2y+z = 2\) and \(x=2y\) and \(x=0\) and \(z=0.\)
    3. The solid over the base region \(R\) in the upper half-plane bound by the circles \(x^2+y^2=1\) and \(x^2+y^2=4\) with ceiling \(z = 3x+4y^2.\)
    4. The solid that lies under the paraboloid \(z=x^2+y^2,\) above the \(xy\)-plane, inside the cylinder \(x^2+y^2=2x.\)
    5. The solid bound above by the surface \({z=x^2+y^2+1}\) and bound below by the region in the \(xy\)-plane bound by the curves \(x=y^2\) and \(x=4.\)
    6. The solid that exists above the base region in the first-quadrant bound by the circle \(x^2+y^2=9\) for \(x \geq 0\) and \(y \geq 0\), and is bound above by the surface \({z = 2x+y.}\)
    7. The solid that exists above the annular base region bound between two circles centered at the origin, one of radius three and the other of radius five, and bound above by the surface \(zx^2 + zy^2 = x^2.\)
    8. The tetrahedron with vertices at the points with coordinates \((0,0,0)\) and \((3,0,0)\) and \((0,6,0)\) and \((0,0,5).\)
    9. The tetrahedron with vertices at the points with coordinates \((0,0,0)\) and \((5,0,0)\) and \((3,7,0)\) and \((0,0,2).\)
    10. The solid bound by the cone \(z^2 = x^2+y^2\) and the cylinder \(x^2+y^2=9.\)
    11. The solid bounded by \(x = z\) and \(x = y\) and \(x = 2-y\) and \(z = 0.\)
    12. The solid bound below by the \(xy\)-plane that exists beneath the paraboloid \({z = 15-3x^2-3y^2.}\)
    13. The expanse inside the sphere centered at the origin of radius five but outside the cylinder centered along the \(z\)-axis of radius four.
    14. The solid bounded by \(x = z^2\) and \(y = z^2\) and \(x = 0\) and \(y = 4.\)
    15. The solid bound by the paraboloid \({z = 1+2x^2+2y^2}\) and all the planes \({z=7}\) and \(x=0\) and \(y=0.\)
    16. The solid enclosed by the parabolic cylinders \(y=1-x^2\) and \(y=x^2-1\) and the planes \(x+y+z=2\) and \(z-2x-2y=10.\)
    17. The solid bound by the paraboloids \(z = 3x^2+3y^2\) and \(z = 4-x^2-y^2.\)
22. Calculate the mass and the center of mass of the triangular lamina with vertices located at \((0,0)\) and \((1,0)\) and \((0,2)\) in the \(xy\)-plane having a density function \(\rho(x,y) = 1+3x+y.\)
23. On a semi-circular lamina, suppose the density at any point is proportional to the distance from that point to the circle’s center. What are the coordinates of the lamina’s center of mass?
24. Determine the center of mass of the region bounded by the parabolas \(x^2=y\) and \(y^2=x\) that has a point-density at \((x,y)\) of \(\sqrt{x}.\)
25. Calculate the moments of inertia about the \(x\)-axis, about the \(y\)-axis, and about the origin, of a unit disk centered at the origin with radius \(r\) and uniform density \(\rho.\)
26. Calculate the surface area of the portion of the surface \(z=x^2+2y+2\) that lies above the triangular region in the \(xy\)-plane with vertices located at the points \((0,0)\) and \((1,0)\) and \((1,1).\)
27. Calculate the surface area of the portion of the paraboloid \(z=x^2+y^2\) that lies under the plane \(z=9.\)
28. Calculate the full surface area of the tetrahedron bound between the coordinate planes and the plane \(3x+y+5z = 4.\)

Problems & Challenges

1. Consider the solid bound below by the square \({[-1,1]\!\times\![-1,1]}\) in the \(xy\)-plane, bound above by a plane with normal vector \(\bm{v}\) that contains the point \({\bigl(0,0,z_0\bigr)}\) and doesn’t intersect the square, and bound around its perimeter by the planes \({x = \pm 1}\) and \({y = \pm 1.}\) Prove that the volume of this solid depends only on \(z_0\) and not on the components of \(\bm{v}.\)
2. James Stewart Consider the region \(R = \big\{(x,y) \mid 1\leq x\leq 3, 2\leq y \leq 5\big\}.\) If \(\lfloor x \rfloor\) denotes the greatest integer in \(x,\) what must the value of the following integral be? \[ \iint\limits_R \lfloor x+y\rfloor \,\mathrm{d}A \]
3. James Stewart Let \(\max\bigl(x^2, y^2\bigr)\) denote the larger of the two numbers \(x^2\) and \(y^2.\) Evaluate \[ \int\limits_{0}^{1} \int\limits_{0}^{1} \mathrm{e}^{\max\bigl(x^2, y^2\bigr)}\,\mathrm{d}y\,\mathrm{d}x \]
4. Consider a disk in the plane with radius \(a.\) Among all the points inside this disk, what is the average distance from a point to the origin?
5. James Stewart What is the average value of \(f(x) = \int_x^1 \cos\bigl(t^2\bigr)\,\mathrm{d}t\) on the interval \([0,1]?\)