Area, Volume, And Cross Section

1. AP Calculus AB/BC FRQ 2011 Question 4 Area, Volume, And Cross Section By Gina Gong

3. (a) Find the area of R. (b) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is rotated about the horizontal line y = 7. (c) Region R is the base of a solid. For each y, where 0 ≤ y ≤ 6, the cross section of the solid taken perpendicular to the y-axis is a rectangle whose height is 3 times the length of its base in region R. Write, but do not evaluate, an integral expression that gives the volume of the solid.

4. (a) Find the area of R. • To find the area of R, first we need to find the limits for the integrand. • As shown in the graph, the area of R begins when x=0 to x= 9

5. (a) Find the area of R (cont) • To find the area of R, we use the rule:

6. (a) Find the Area of R (Cont) • To find the area of R numerically/analytically:

7. (a) Find the area of R (Cont) • You can also use a graphing calculator to find the area of R. Plug in the function into y = To evaluate the integrand, press 2nd Trace (Calc) and then choose the integral function #7. It will then ask you for the lower limit. As stated earlier the lower limit is when x = 0. When rounding y = 17.999968 we get 18. The area of R is equal to 18. The upper limit it when x = 9. The calculator will then give you the answer for the area under the curve. We will round this to the nearest three digits.

8. (b) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is rotated about the horizontal line y = 7. • To find the volume of the solid generated we use the formula: • Because it is rotated about the horizontal line y=7 we subtract the functions from the line y=7 Therefore we get Simplified we get We also know that the limit is from x=0 to x=9 (from the graph) NOTE: This is a NO CALCULATOR question, therefore you do not need to solve for the volume. Also the question states WRITE but do not evaluate, an integral expression.

9. (c) Region R is the base of a solid. For each y, where 0 ≤ y ≤ 6, the cross section of the solid taken perpendicular to the y-axis is a rectangle whose height is 3 times the length of its base in region R. Write, but do not evaluate, an integral expression that gives the volume of the solid. • This question asks us to write an equation for the volume of the cross section for y where 0 ≤ y ≤ 6. • Therefore we have to solve for x • Also because we know that it is the cross section perpendicular to the y-axis, we know that the limit (given) for the integrand is from y=0 to y=6 • The Area of a Rectangle is equal to is Length x Width

10. (c) Region R is the base of a solid. For each y, where 0 ≤ y ≤ 6, the cross section of the solid taken perpendicular to the y-axis is a rectangle whose height is 3 times the length of its base in region R. Write, but do not evaluate, an integral expression that gives the volume of the solid. • The information given states that region R is the base of the solid and the cross section of the solid is a rectangle whose HEIGHT is three times the length of its base in region R. NOTE: This is a NO CALCULATOR question, therefore you do not need to solve for the volume. Also the question states WRITE but do not evaluate, an integral expression.

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