math CALCULUS

Math 252 – Homework #4 Name ________________________ Due: Wednesday July 24, 2019

1) The rate of energy consumed by an appliance over 𝑡 hours, is given by 𝐸′(𝑡) (measured in joules/hr), where 𝐸(𝑡) is the energy consumed (measured in joules) by the appliance.

𝐸′(𝑡) = 2 + sin ( 𝜋

12 ⋅ 𝑡) 𝑗𝑜𝑢𝑙𝑒𝑠/ℎ𝑟

a) Find and graph the power consumed by the appliance 𝐸(𝑡) assuming 𝐸(0) = 2 𝑗𝑜𝑢𝑙𝑒𝑠 b) How much energy is consumed during a one-day period?

2) First, sketch the region, R, bounded by the curves 𝑦 = 2𝑥 and 𝑦 = 𝑥 2 − 2𝑥 in the xy- plane. Sketch a sample Riemann rectangular slice in R to guide your solution process. Then, set up the integral to determine the area of that region. Finally, evaluate the integral to determine the area of the region.

3) Use the general slicing method to find the volume of the following solid.

The solid whose base is the region bounded by the curves 𝑦 = 𝑥 2 and𝑦 = 2 − 𝑥 2, and whose cross-sections through the solid perpendicular to the x-axis are squares.

4) Use the washer method to determine the volume of the object described below.

Determine the volume of the object created by rotating the region bounded by

𝑦 = 1, 𝑦 = √cos(𝑥) and the x-axis on the interval - p

2 , p

2

é

ëê ù

ûú , about the x-axis.

5) Use the diagrams below to answer the following questions:

6) Explain why someone who memorizes the washer and shell method formulas in the text should be extremely careful when solving problems of revolution.

a) Suppose the bounded region in the graph to the right is rotated about the y-axis. Which integration method will require only one integral?

(Circle one)

Washer Method Shell Method

Draw in an appropriate rectangular slice in the diagram for the method you chose.

b) Suppose the bounded region in the graph to the right is rotated about the x-axis. Which integration method will require only one integral? (Circle one)

Washer Method Shell Method

Draw in an appropriate rectangular slice in the diagram for the method you chose.

7) Set up and evaluate an integral to find the volume of the solid obtained by rotating the given region bounded by 𝑦 = 𝑥 2 and 𝑥 = 𝑦2

a) About the line 𝑥 = −1, using the washer method!

b) About the line 𝑥 = −1 using the shell method!

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