SUNY Geneseo Department of Mathematics
Math 223 01
Fall 2022
Prof. Doug Baldwin
Complete by Sunday, November 27
Grade by Friday, December 2
This exercise reinforces your understanding of definite integrals of multivariable functions and of scalar line integrals. It contributes to the following learning outcomes for this course:
This exercise is based on material in sections 4.3 and 5.2 in our textbook. We covered those sections in classes between November 7 and 14.
Solve each of the following problems.
Exercise 18 in section 4.3E of the textbook:
Integrate \(f(x,y) = x^2 + xy\) over the polar region defined by \(1 \le r \le 2\) and \(\pi \le \Theta \le 2\pi\).
Exercise 26 in section 4.3E of the textbook:
Show that the following equation holds between an integral in rectangular coordinates and a (supposedly) equivalent one in polar coordinates. Then evaluate the integral in whichever coordinate system you think will be easiest.
\[\int_1^2 \int_0^x x^2+y^2\,dy\,dx = \int_0^\frac{\pi}{4} \int_{\sec\Theta}^{2\sec\Theta} r^3\,dr\,d\Theta\]Find the value of
\[\int_C xy^4\,ds\]where \(C\) is the right half of the circle \(x^2+y^2=16\).
Find the value of
\[\int_C x + \sqrt{y} - z^2\,ds\]where \(C\) is the 2-part path that follows a straight line from point \((0,0,0)\) to point \((1,0,0)\), and then from there to point \((1,1,1)\).
I will grade this exercise during an individual meeting with you. That meeting should happen on or before the “Grade By” date above. During the meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Sign up for the meeting via Google calendar. Please have a written solution to the exercise ready to share with me during your meeting, as that will speed the process along.