For each triangle,

(a) Find an integral expression for the area of the triangle using the slice and variable of integration indicated.

(b) Evaluate your integral and check that it agrees with the area you find using the formula

Answer

(a) This is similar to the example and the first problem. However, instead of measuring the distance from the point of the triangle to the slice, *y* measures the distance from the base of the triangle to the slice.

We will call the length of a slice *x*. When we use similar triangles to find *x*, we have to make sure to write 8 – *y* for the height of the smaller triangle. We know that

so

The area of the slice at height *y* is

Since *y* ranges from 0 (at the bottom of the triangle) to 8 (at the top of the triangle) the area of the triangle is

(b) Using the integral we get

Using the formula we get