Question

In a right sum with n = 4 rectangles for the area between the graph of $f\left(x\right)=x^2-1$and the x-axis on [0, 2], how many rectangles have negative heights? What if n = 7? In each case, is the estimate for the signed area positive or negative? Is the exact value of the signed area positive or negative?

Short answer

The number of rectangles with negative height is 2. The exact value of signed area is positive.

Step-by-step solution

Step 1. Given information

$f\left(x\right)=x^2-1$in the interval [0,2]

Step 2. Determining the negative rectangles

Plotting the graph of $x^2-1$,

Therefore, the number of rectangles with negative height is 2.

Step 3.  Calculating the area

${\int }_0^2{x}^2-1dx=\frac{1}{3}\left(2^3\right)-1\left(2\right)$

${\int }_0^2{x}^2-1dx=\frac{8}{3}-2$

${\int }_0^2{x}^2-1dx=\frac{8-6}{3}$

${\int }_0^2{x}^2-1dx=\frac{2}{3}$

Hence, exact value of signed area is positive.