Definite Integrals Exercises

Answer

First we need to know what  is. When we look at the graph, we see a trapezoid with heights 6 and 12 and width 2.

The area of this trapezoid is

In order for  to be 18 also, m must be 9.

Answer

Here's the graph of f (x) = x on [-2, 4]:

From this graph we can see that

We want

m(4 – (-2)) = 6.

Since (4 – (-2)) = 6, We must have m = 1. The weighted area between f (x) = x and the x-axis on [-2,4] is the same as the weighted area between m = 1 and the x-axis on [-2, 4].

Answer

The average value of f (x) = x on [-10, 0] is

Since

the average value of f (x) = x on [-10, 0] is

Since f (x) is below the x-axis on all of [-2, 2], it's reassuring that we got a negative number for the average value.

Answer

The function  on [-2, 2] describes half a circle.

The area of this half-circle is

so

The average of f on [-2, 2] is

Since f (x) is below the x-axis on all of [-2, 2], it's reassuring that we got a negative number for the average value.

Answer

The average value of f (x) = sin x on [-π, π] is

Since sine is an odd function, its integral on [-π, π] is 0. So the average value of f (x) = sin x on [-π, π] is

.

Answer

The graph of f (x) = 3x + 2 on [1, 4] describes a trapezoid with width 3 and heights 5 and 14.

The area of this trapezoid is

so

The average of f on [1,4] is

.

Answer

The line -2x – 1 hits the x-axis when 

so the triangle above the x-axis has base 3.5 and the triangle below the x-axis has base 2.5.

Then

We can conclude that the average of f on [-4, 2] is