CALCULUS II

Question

Set up the definite integral that gives the area of the region.
y1 = (x − 7)3
y2 = x − 7

Answer 1

First find the bounds.  This can be found by setting y₁ equal to y₂.

 

(x - 7)³ = x - 7

 

(x - 7)³ - x + 7 = 0

 

(x - 7)³ - (x - 7) = 0

 

(x - 7)((x - 7)² - 1) = 0

 

(x - 7)(x² - 14x + 49 - 1) = 0

 

(x - 7)(x² - 14x + 48) = 0

 

(x - 7)(x - 6)(x - 8) = 0

 

x = 6         x = 7            x = 8

 

You have 3 bounds.  This means that you will have 2 subareas under the curve for that whole area.  You will add two definite integrals.

 

⁷∫ ₆ [(x - 7)³ - (x - 7)]dx  +  ⁸∫ ₇ [(x - 7) - (x - 7)³]dx

 

 

To make it easier for yourself, find the integral one piece at a time.