1. A circle, centered at the origin with radius r, is shown below along with the equation for the top half of the circle. Whole circle: x2+ 312: r2 Top half of circle: y = \\/ r2 x2 a. Suppose the top half of the circle is rotated about the x-axis to create a sphere. Setup at r3 and solve a definite integral that derives the formula for volume of a sphere (using the 3 disk, washer, or shell method). b. Suppose the whole circle is graphed. Setup and solve a definite integral that derives the C = 2 31- r formula for circumference of a circle (using the arc length formula). c. Suppose the top half ofthe circle is rotated about the x-axis to create a sphere. Setup 2 and solve a definite integral that derives the formula for surface area of a sphere (using A = 4 .71" r the surface area formula). 2. A thin plate is described (and enclosed) by the region below. Region: y=x2+ 1, x= 2, x=2, x axis a. Where is the center of mass when the density is constant? b. Where is the center of mass when the density is variable? 5( I) =x + 1 Reminder: Complete all work in your handwriting Sketch ALL two-dimensional graphs by hand and fully label each one Show ALL calculus/algebra work Give exact answers (do NOT give decimal approximations) Grading Rubric for each part: Strong (3 points) Moderate (2 points) Weak (1 points) Strategy Identifies all required ideas Identifies most of the required Identifies some of the required for solving the problem ideas for solving the problem ideas for solving the problem Organization Very organized and easy Somewhat organized Disorganized to follow Notation Uses all the required Uses most of the required Uses some of the required notation notation notation Conclusion Interprets the results Interprets the results correctly Interprets the results incorrectly correctly with a thorough and/or gives a correct and/or gives an incorrect conclusion conclusion conclusion