Section Practice
¶These are practice problems for the tests. Solutions are in the next section. While we will not present these, I am happy to answer questions about them in class.
Basics
Evaluate \(\dsp \int x^3y-3x \ dx\) and \(\dsp \int_1^3 x^3y-3x \ dy\)
Evaluate \(\dsp \int xye^{xy} \ dx\) and \(\dsp \int_{-1}^1 xye^{xy} \ dy\)
Compute the area of the region bounded by the parabola \(y=x^{2}-2\) and the line \(y=x\) by first computing a single integral with respect to \(x\) and then computing a single integral with respect to \(y.\)
Compute the area bounded by \(y=x^3\) and \(y=5x\) in four ways. (a) Single integral with respect to \(x,\) (b) single integral w.r.t. \(y,\) (c) double integral, \(dx \ dy,\) and (d) double integral, \(dy \ dx.\)
Regions
Sketch the region \(\phi= \pi/6\) in spherical coordinates.
Sketch the region bounded by \(r=1\) and \(r=2\sin(\theta)\) in polar coordinates.
Sketch the region bounded by \(x^{2}+y^{2}\leq 9\) and write in polar coordinates.
Sketch the region between \(x^{2}+y^{2}=25\text{,}\) \(x^{2}+y^{2}=4\text{,}\) and \(x\geq 0\) and write in polar coordinates.
Find the area of the region \(D\) bounded by \(y= \cos(x)\) and \(y= \sin(x)\) on the interval \(\dsp \Big[0,\frac{\pi }{4} \Big]\text{.}\)
Integration
Compute \(\dsp \int_0^2 \int_0^3 x^2y^2-3xy^5\ dy \ dx\) and \(\dsp \int_0^3 \int_0^2 x^2y^2-3xy^5\ dx \ dy.\) Are they equal? What theorem is this an example of?
Evaluate \(\dsp \int_{0}^{1}\int_{0}^{y} e^{y^{2}} \ dx \ dy\text{.}\)
Compute \(\dsp \int_{R}x^{2}e^{xy}dA\) where \(R = \{(x, y): 0\leq x\leq 3, 0\leq y\leq 2 \}\text{.}\)
Compute \(\dsp \int_{0}^{3} \int_{0}^{2} \sqrt{2x+y} \ dy \ dx\text{.}\)
Compute \(\dsp \int_{R}\frac{\ln\sqrt{y}}{xy}dA\) where \(R = \{(x, y): 1\leq x\leq 4, 1\leq y\leq e \}\text{.}\)
Evaluate \(\dsp \int_{D} 160xy^{3} \ dA\) where \(D\) is the region bounded by \(y=x^{2}\) and \(y=\sqrt{x}\text{.}\)
Sketch and find the volume of the solid bounded above by the plane \(z=y\) and below in the \(xy\)-plane by the part of the disk \(x^{2}+y^{2}\leq 1\text{.}\)
Sketch the region, \(D\text{,}\) that is bounded by \(x=y^{2}\) and \(x=3-2y^{2}\) and evaluate \(\int_{D} (y^{2}-x)dA\)
Compute \(\dsp \int_{0}^{2}\int_{0}^{\sin(x)} y \cos(x) \ dy \ dx\text{.}\)
Determine the endpoints of integration for \(\dsp \iint_S e^{xy} dA\) where \(S\) is the region bounded by \(y= \sqrt x\) and \(\dsp y={x \over 9}\text{.}\) Don't integrate.
Determine the endpoints of integration for \(\dsp \iint_S dA\) where \(S\) is bounded by a the \(x=y^{2}+4y\) and \(x=3y+2\text{.}\)
Determine the endpoints of integration for \(\dsp \iint_{S} 2x \ dA\) where \(S\) is the region bounded by \(yx^{2}=1\text{,}\) \(y=x\text{,}\) \(x=2\text{,}\) and \(y=0\text{.}\)
Evaluate \(\dsp \iint _D dA\) where \(D\) is the region bounded by the ellipse \(\dsp \frac{x^{2}}{4}+\frac{y^{2}}{9}=1\)
Polar and Spherical Coordinate Integration
Sketch the region \(D\) between \(r= \dsp \cos \Big(\frac{\theta }{2} \Big)\) and \(x^{2}+y^{2}=1\) with \(0\leq \theta \leq \pi\text{.}\) Evaluate \(\dsp \int_{D} 1 dA\text{.}\)
Consider \(\dsp \int_{R} x^{2}+y^{2}+1 \ dA\) where \(R = \{(x, y): x \ge 0, 9 \le x^{2}+y^{2} \le 16 \}\text{.}\) Write the integral in both rectangular and polar coordinates. Compute each to verify your answer.
Find the volume of the solid bounded by the cone \(\dsp \phi = {{\pi} \over 6}\) and the sphere \(\rho=4.\)
Coordinate Transformations
Find the Jacobian for the transformation: \(x=u^{2}+v^{2}+w\text{,}\) \(y=uw-v\text{,}\) and \(\dsp z=\ln(w)-\frac{v}{u}\)
Evaluate \(\dsp \int_{0}^{4} \int_{y/2}^{(y/2)+1} \frac{2x-y}{2} \ dx \ dy\) using \(u= \dsp \frac{2x-y}{2}\) and \(v= \dsp \frac{y}{2}\text{.}\)
Use the transformation \(x= \dsp \frac{u}{v}\) and \(y=v\) to rewrite (but not evaluate) the double integral \(\int \int \sqrt{xy^{3}} \ dx \ dy\) over the region in the plane bounded by the \(x\)-axis, the \(y\)-axis, and the lines \(y=-2x+2\) and \(x+y=7\text{.}\)
Compute \(\dsp \int_0^1 \int_{0}^{y^2} (1-y) \sin \Big({x \over y} \Big) \ dx \ dy\) using \(\dsp u = {x \over y}\) and \(v = 1-y\text{.}\)
Write an integral in rectangular coordinates that represents the area enclosed by the ellipse \(\dsp \frac{x^{2}}{16}+\frac{y^{2}}{49}=1.\) Now, compute this integral by using the transformation \(x=4u\) and \(y=7v\text{.}\)
Find the volume of the ellipsoid \(\dsp {{x^2} \over {a^2}} + {{y^2} \over {b^2}} + {{z^2} \over {c^2}} = 1\) using the transformation \(x=au\text{,}\) \(y=bv\text{,}\) \(z=cw\text{.}\)
Evaluate the triple integral \(\dsp \int_0^6 \int_0^8 \int_{y \over 2}^{{y \over 2}+4} {{2x-y} \over 2} + {y \over 6} + {z \over 3} dx\ dy\ dz\) using the transformation \(\dsp u={{2x-y} \over 2}\text{,}\) \(\dsp v={y \over 2}\text{,}\) and \(\dsp w = {z \over 3}\text{.}\)
Triple Integrals, Cylindrical, and Spherical Coordinates
Sketch and find the volume of the solid formed by \(f(x,y)=4x+2y\) above the region in the \(xy\)-plane bounded by \(x=2\text{,}\) \(x=4\text{,}\) \(y=-x\text{,}\) \(y=x^2\text{.}\)
Fill in the blanks: \(\dsp{\int_0^1{\int_0^\frac{1-x}{2} \int_0^{1-x-2y}{f(x,y,z)\ dz\ dy\ dx}}}\\ =\int_\_^\_{\int_\_^\_{\int_\_^\_ f(x,y,z) \ {dy\ dz\ dx}}}\\ =\int_\_^\_{\int_\_^\_{\int_\_^\_ f(x,y,z) \ {dy\ dx\ dz}}}\\ =\int_\_^\_{\int_\_^\_{\int_\_^\_ f(x,y,z) \ {dx\ dy\ dz}}}\)
\(\dsp \int_0^6 \int_0^{{12-2x} \over 3} \int_0^{{12-2x-3y} \over 4} f \ dz\ dy\ dx = \int_{\underline{\ }}^{\underline{\ }} \int_{\underline{\ }}^{\underline{\ }} \int_{\underline{\ }}^{\underline{\ }} f \ dy\ dx\ dz = \int_{\underline{\ }}^{\underline{\ }} \int_{\underline{\ }}^{\underline{\ }} \int_{\underline{\ }}^{\underline{\ }} f \ dx\ dz\ dy\)
\(\dsp \int_0^\frac{1}{6} \int_0^{{\sqrt{1-36x^2}} \over 3} \int_0^{{\sqrt{1-36x^2-9y^2}} \over 2} f \ dz\ dy\ dx = \int_{\underline{\ }}^{\underline{\ }} \int_{\underline{\ }}^{\underline{\ }} \int_{\underline{\ }}^{\underline{\ }} f \ dx\ dz\ dy\)
Find the volume of the solid bounded by \(x^2+y^2+z=8\) and \(z=4\text{.}\)
Evaluate \(\dsp \int_0^2 \int_0^{\sqrt{4-x^2}} \int_0^{\sqrt{4-x^2-y^2}} z \sqrt{4-x^2 - y^2} \ dz\ dy\ dx\) by converting to (a) cylindrical coordinates; (b) spherical coordinates.
Application - Center of Mass
Suppose \(\delta(x,y)=x+y\) is the density function of a thin sheet of material bounded by the curve \(x^2=4y\) and \(x+y=8\text{.}\) Find its total mass. Find its first moments. Find its center of mass. Find its second moments.