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Multivariable Calculus : Double Integrations on Plane

Study concepts, example questions & explanations for Multivariable Calculus

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Example Questions

Example Question #1 : Change Of Variables

Convert the following into spherical coordinates.

$x=\sin(t)$

$y=\cos(t)$

z=1

Possible Answers:

$\rho=\sqrt{2}$

$\phi=0$

$\theta=\tan(\cot(t))$

$\rho=\sqrt{2}$

$\phi=\frac{\pi}{4}$

$\theta=\tan^{-1}(\cot(t))$

$\rho=1$

$\phi=\frac{\pi}{2}$

$\theta=\tan(\cot(t))$

$\rho=\sqrt{2}$

$\phi=\frac{\pi}{4}$

$\theta=\tan^{-1}(\cot(t))$

$\rho=2$

$\phi=-\frac{\pi}{4}$

$\theta=\tan^{-1}(\cot(t))$

Correct answer:

$\rho=\sqrt{2}$

$\phi=\frac{\pi}{4}$

$\theta=\tan^{-1}(\cot(t))$

Explanation:

In order to convert to spherical coordinates , we need to remember the conversion equations.

$\rho=\sqrt{x^2+y^2+z^2}$

$\theta=\tan^{-1}\Big(\frac{y}{x}\Big)$

$\phi=\tan^{-1}\Big(\frac{\sqrt{x^2+y^2}}{z}\Big)$

Now lets apply this to our problem.

$\rho=\sqrt{(\sin(t))^2+(\cos(t))^2+1^2}=\sqrt{1+1}=\sqrt{2}$

$\phi=\tan^{-1}\Big(\frac{\sqrt{(\sin{t})^2+(\cos{t})^2}}{1}\Big)=\tan^{-1}(1)=\frac{\pi}{4}$

$\theta=\tan^{-1}\Big(\frac{\cos(t)}{\sin(t)}\Big)=\tan^{-1}(\cot(t))$

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Example Question #2 : Change Of Variables

Evaluate $\int \int_R 3x+2y \ dA$ , where is the trapezoidal region with vertices given by (0,0) , (5,0) , $(\frac{5}{2}, \frac{5}{2})$ , and $(\frac{5}{2}, -\frac{5}{2})$ ,

using the transformation x=2u+3v , and y=2u-3v .

Possible Answers:

$\frac{35}{4}$

$\frac{37}{4}$

$\frac{375}{2}$

$\frac{375}{4}$

Correct answer:

$\frac{375}{4}$

Explanation:

The first thing we have to do is figure out the general equations for the lines that create the trapezoid.

y=x

y=-x

y=-x+5

y=x-5

Now we have the general equations for out trapezoid, now we need to plug in our transformations into these equations.

y=x

2u+3v=2u-3v

$-6v=0\rightarrow v=0$

y=-x

2u+3v=-2u+3v

$4u=0\rightarrow u=0$

y=-x+5

2u+3v=-2u+3v+5

$4u=5\rightarrow u=\frac{5}{4}$

y=x-5

2u+3v=2u-3v-5

$-6v=-5\rightarrow v=\frac{5}{6}$

So our region is a rectangle given by $0 \leq u \leq \frac{5}{4}$ , $0 \leq v \leq \frac{5}{6}$

Next we need to calculate the Jacobian.

$\begin{vmatrix} 2 & 3\\ 2 & -3 \end{vmatrix} =-6-6=-12$

Now we can put the integral together.

$\int_{0}^{\frac{5}{6}} \int_{0}^{\frac{5}{4}} (3(2u+3v)+2(2u-3v))) \cdot \left | -12\right | \ du\ dv$

$=\int_{0}^{\frac{5}{6}} \int_{0}^{\frac{5}{4}} 120u+36v \ du\ dv$

$=\int_{0}^{\frac{5}{6}} 60u^2+36uv \Big|_{0}^{\frac{5}{4}}\ dv$

$=\int_{0}^{\frac{5}{6}} 60( \frac{5}{4})^2+36(\frac{5}{4})v-0 \ dv$

$=\int_{0}^{\frac{5}{6}} \frac{375}{4}+45v \ dv$

$=\frac{375}{4}v+\frac{45}{2}v^2\Big|_{0}^{\frac{5}{6}} \ dv$

$=\frac{375}{4}\cdot\frac{5}{6}+\frac{45}{2}\cdot\Big(\frac{5}{6}\Big)^2 -0=\frac{375}{4}$

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Example Question #1 : Change Of Variables

Convert the following into spherical coordinates.

$x=\sin(t)$

$y=\cos(t)$

z=1

Possible Answers:

$\rho=1$
$\phi=\frac{\pi}{2}$
$\theta=\tan(\cot(t))$

$\rho=\sqrt{2}$
$\phi=-\frac{\pi}{4}$
$\theta=\tan^{-1}(\cot(t))$

$\rho=\sqrt{2}$
$\phi=0$
$\theta=\tan(\cot(t))$

$\rho=2$
$\phi=-\frac{\pi}{4}$
$\theta=\tan^{-1}(\cot(t))$

$\rho=\sqrt{2}$
$\phi=\frac{\pi}{4}$
$\theta=\tan^{-1}(\cot(t))$

Correct answer:

$\rho=\sqrt{2}$
$\phi=\frac{\pi}{4}$
$\theta=\tan^{-1}(\cot(t))$

Explanation:

In order to convert to spherical coordinates , we need to remember the conversion equations.

$\rho=\sqrt{x^2+y^2+z^2}$

$\theta=\tan^{-1}\Big(\frac{y}{x}\Big)$

$\phi=\tan^{-1}\Big(\frac{\sqrt{x^2+y^2}}{z}\Big)$

Now lets apply this to our problem.

$\rho=\sqrt{(\sin(t))^2+(\cos(t))^2+1^2}=\sqrt{1+1}=\sqrt{2}$

$\phi=\tan^{-1}\Big(\frac{\sqrt{(\sin{t})^2+(\cos{t})^2}}{1}\Big)=\tan^{-1}(1)=\frac{\pi}{4}$

$\theta=\tan^{-1}\Big(\frac{\cos(t)}{\sin(t)}\Big)=\tan^{-1}(\cot(t))$

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Example Question #1 : Double Integration

Calculate the following Integral.

$\int_{0}^{1}\int_{x}^{x^2} xy^2 dydx$

Possible Answers:

$-\frac{1}{120}$

120

$-\frac{3}{120}$

$\frac{1}{120}$

Correct answer:

$-\frac{3}{120}$

Explanation:

$\int_{0}^{1}\int_{x}^{x^2} xy^2 dydx=\int_{0}^{1}\Big(\int_{x}^{x^2} xy^2 dy\Big)dx$

Lets deal with the inner integral first.

$\int_{x}^{x^2}xy^2 dy$

$\int_{x}^{x^2}xy^2 dy=\frac{1}{3}y^3x\Big|_{x}^{x^2}$

$=(\frac{1}{3}(x^2)^3\cdot x)-(\frac{1}{3}x^3\cdot x)$

$=\frac{1}{3}x^7-\frac{1}{3}x^4$

Now we evaluate this expression in the outer integral.

$\int_{0}^{1} \frac{1}{3}x^7-\frac{1}{3}x^4dx$

$=\frac{1}{3}\int_{0}^{1} x^7-x^4dx$

$=\frac{1}{3}\Big( \frac{1}{8}x^8-\frac{1}{5}x^5\Big)\Big|_{0}^{1}$

$=\frac{1}{3}\Big(\frac{1}{8}1^8-\frac{1}{5}1^5\Big)-\frac{1}{3}\Big(\frac{1}{8}0^8-\frac{1}{5}0^5\Big)$

$=\frac{1}{3}\Big(\frac{1}{8}-\frac{1}{5}\Big)-\frac{1}{3}\Big(0-0\Big)$

$=\frac{1}{3}\Big(\frac{5}{40}-\frac{8}{40}\Big)$

$=\frac{1}{3}\Big(-\frac{3}{40}\Big)=-\frac{3}{120}$

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Example Question #1 : Double Integrations On Plane

Calculate the following Integral.

$\int_{0}^{1}\int_{x}^{x^2} xy^2 dydx$

Possible Answers:

$\frac{1}{120}$

$-\frac{3}{120}$

$-\frac{1}{120}$

120

Correct answer:

$-\frac{3}{120}$

Explanation:

$\int_{0}^{1}\int_{x}^{x^2} xy^2 dydx=\int_{0}^{1}\Big(\int_{x}^{x^2} xy^2 dy\Big)dx$

Lets deal with the inner integral first.

$\int_{x}^{x^2}xy^2 dy$

$\int_{x}^{x^2}xy^2 dy=\frac{1}{3}y^3x\Big|_{x}^{x^2}$

$=(\frac{1}{3}(x^2)^3\cdot x)-(\frac{1}{3}x^3\cdot x)$

$=\frac{1}{3}x^7-\frac{1}{3}x^4$

Now we evaluate this expression in the outer integral.

$\int_{0}^{1} \frac{1}{3}x^7-\frac{1}{3}x^4dx$

$=\frac{1}{3}\int_{0}^{1} x^7-x^4dx$

$=\frac{1}{3}\Big( \frac{1}{8}x^8-\frac{1}{5}x^5\Big)\Big|_{0}^{1}$

$=\frac{1}{3}\Big(\frac{1}{8}1^8-\frac{1}{5}1^5\Big)-\frac{1}{3}\Big(\frac{1}{8}0^8-\frac{1}{5}0^5\Big)$

$=\frac{1}{3}\Big(\frac{1}{8}-\frac{1}{5}\Big)-\frac{1}{3}\Big(0-0\Big)$

$=\frac{1}{3}\Big(\frac{5}{40}-\frac{8}{40}\Big)$

$=\frac{1}{3}\Big(-\frac{3}{40}\Big)=-\frac{3}{120}$

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Multivariable Calculus : Double Integrations on Plane

Study concepts, example questions & explanations for Multivariable Calculus

All Multivariable Calculus Resources

Example Questions

Example Question #1 : Change Of Variables

Example Question #2 : Change Of Variables

Example Question #1 : Change Of Variables

Example Question #1 : Double Integration

Example Question #1 : Double Integrations On Plane

All Multivariable Calculus Resources

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