- #1
songoku
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- Homework Statement
- Let R be the region which lies inside the circle ##x^2+y^2=100## and outside the circle ##x^2-10x+y^2=0##
a) Sketch the two circles and shade the region R in your diagram
b) Evaluate the integral ##\iint_R \frac{1}{\sqrt{x^2+y^2}}~dA##
- Relevant Equations
- ##\iint_R f(x,y)~dA=\int_{\alpha}^{\beta} \int^{b}_{a} f(r,\theta) r~dr~d\theta##
##r=2a \cos\theta+2b\sin\theta##
The polar form of ##x^2+y^2=100## is ##r=10## and polar form of ##x^2-10x+y^2=0## is ##r=10 \cos\theta##
My idea is to divide the working into two parts:
1) find the integral in 1st quadrant and multiply by 2 to include the region in 4th quadrant
2) find the integral in 2nd quadrant and multiply by 2 to include the region in 3rd quadrant
Integral in 1st quadrant:
$$\int_{0}^{\frac{\pi}{2}} \int^{10}_{10\cos\theta} \frac{1}{r}r~dr~d\theta$$
$$=\int_{0}^{\frac{\pi}{2}} \int^{10}_{10\cos\theta} dr~d\theta$$
Integral in 2nd quadrant:
$$\int_{\frac{\pi}{2}}^{\pi} \int^{10}_{0} dr~d\theta$$
Am I correct? Thanks
My idea is to divide the working into two parts:
1) find the integral in 1st quadrant and multiply by 2 to include the region in 4th quadrant
2) find the integral in 2nd quadrant and multiply by 2 to include the region in 3rd quadrant
Integral in 1st quadrant:
$$\int_{0}^{\frac{\pi}{2}} \int^{10}_{10\cos\theta} \frac{1}{r}r~dr~d\theta$$
$$=\int_{0}^{\frac{\pi}{2}} \int^{10}_{10\cos\theta} dr~d\theta$$
Integral in 2nd quadrant:
$$\int_{\frac{\pi}{2}}^{\pi} \int^{10}_{0} dr~d\theta$$
Am I correct? Thanks