Q. 14 - Calculus

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?

Step-by-Step Solution

Answer

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

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.