Q. 11 - Calculus

Question

Consider the region between f and the x-axis on$[-2, 4]$ as in the graph next at the left.

Part (a): Draw the rectangles of the left-sum approximation for the area of this region, with $n = 8$.

Part (b): Express the signed area.

Part (c): The absolute area of the region with definite integrals that do not involve absolute values.

Step-by-Step Solution

Answer

Part (a): The required rectangle is given below,

Part (b): On expressing the signed area, we get, ${\int }_{-2}^4 f\left(x\right) dx$.

Part (c): On writing the absolute area of the region with definite integrals, we get, ${\int }_{-2}^{-1}f\left(x\right) dx-{\int }_{-1}^{3}f\left(x\right) dx+{\int }_3^{4}f\left(x\right) dx$.

Part (a) Step 1. Given information.

Consider the given question,

$[-2, 4]$

Also,

Part (a) Step 1. Draw the mentioned region.

The rectangles are shown below with $n=8$,

Part (b) Step 1. Expressing the signed area.

Write the signed area,

$={\int }_{-2}^4 f\left(x\right) dx$

Part (b) Step 1. Express the absolute area.

Write the absolute area,

$={\int }_{-2}^{-1}f\left(x\right) dx-{\int }_{-1}^{3}f\left(x\right) dx+{\int }_3^{4}f\left(x\right) dx$