Chapter 11.3, Problem 7E

To determine

Find the area of the curve $y=x^2$ from $x=-3\text{to}x=3$ and above the x − axis.

Answer to Problem 7E

The approximate area under the curve is 17.5 sq units.

Explanation of Solution

Given:

Find the area of the curve $y=x^2$ from $x=-3\text{to}x=3$ and above the x − axis.

Concept Used:

It is like sum of areas of 4 rectangles

The graph: $y=x^2$

  

Calculation:

It is like sum of areas of 6 rectangles.

We find the area under the curve $y=x^2$ from $x=-3\text{to}x=3$ and above the x − axis.

According to the graph there are six rectangles.

First rectangle from $x=-3\text{to}x=-2$

Second rectangle from $x=-2\text{to}x=-1$

Third rectangle from $x=-1\text{to}x=0$

Fourth rectangle from $x=0\text{to}x=1$

Fifth rectangle from $x=1\text{to}x=2$

Sixth rectangle from $x=2\text{to}x=3$

Width of each rectangle is 1 unit.

We use the midpoint of each rectangle.

Now find the height of each rectangle,

So height of first rectangle would be $x=-2.5\Rightarrow {(-2.5)}^2=6.25$

Height of second rectangle would be $x=-1.5\Rightarrow {(-1.5)}^2=2.25$

Height of third rectangle would be $x=-0.5\Rightarrow {(-0.5)}^2=0.25$

Height of fourth rectangle would be $x=0.5\Rightarrow {(0.5)}^2=0.25$

Height of fifth rectangle would be $x=1.5\Rightarrow {(1.5)}^2=2.25$

Height of sixth rectangle would be $x=2.5\Rightarrow {(2.5)}^2=6.25$

The area of 1^st, 2^nd, 3^rd and 4^th, 5^th, 6^th rectangles are: $6.25,2.25,0.25,0.25,2.25,6.25$ sq unit.

So the approximate area under the curve = sum of all 6 rectangles

So the approximate area under the curve: $6.25+2.25+0.25+0.25+2.25+6.25=17.5$ sq unit.

Thus, the approximate area under the curve is 17.5 sq units.