.
- B The formula for the area under a curve using right-endpoint rectangles is: A = (y 1 + y 2
+ y 3 +...+ yn), where a and b are the x-values that bound the area and n is the number of
rectangles. Since we are interested in the right-endpoints, the x-coordinates are x 1 = , x 2 = ,
x 3 = , and x 4 = 2. The y-coordinates are found by plugging these values into the equation for
y, so y 1 = 3.5625, y 2 = 4.25, y 3 = 5.0625, and y 4 = 6. Then, A = (3.5625 + 4.25 +
5.0625 + 6) = 4.71875.
- C The formula for the area under a curve using inscribed trapezoids is: A = (y 0 +
2 y 1 + 2yn−1 + yn, where a and b are the x-values that bound the area and n is the number of
rectangles. The x-coordinates are x 0 = 1, x 1 = , x 2 = ,x 3 = , and x 4 = 2. The y-coordinates
are found by plugging these values into the equation for y, so y 0 = 3, y 1 = 3.5625 y 2 = 4.25 y 3 =
5.0625, and y 4 = 6. Then, A = (3 + 2(3.5625) + 2(4.25) + 2(5.0625) + 6) =
4.34375.
- D Use the Fundamental Theorem of Calculus: f (x) dx = F (b) − F (a) and u-substitution. For
this problem, u = x^2 and du = 2x dx. Then:
dx = du = = − = 0
- D The volume of a sphere is V = π R 3 . Using differentials, the change will be dV = 4πR^2 dR.
Substitute in R = 10 and dR = 0.02 to get
dV = 4π(10^2 )(0.02)
dV = 8π ≈ 25.133 cm^3
