Area, Volume, and Arc Length Exercises

Answer

(a) This problem is almost exactly like the example. Let x be the length of the slice at distance h from the top of the triangle. The blue and pink triangles are similar:

This means

Multiplying both sides by h,

The area of the slice at depth h from the tip of the triangle is

Since h ranges from 0 at the top of the triangle to 6 at the bottom of the triangle, the area of the triangle is

(b) This integral works out to

This agrees with the area we get using the area formula:

Answer

(a) The variable h measures the distance from the right endpoint of the triangle to the slice. When h = 0 the slice is at the right tip of the triangle, and when h = 10 the slice is all the way on the left.

We'll call the height of a slice y. Now we're perfectly set up with some similar triangles, so we can get a ratio that relates y and h:

so

The area of the slice at distance h from the right tip of the triangle is

The limits of integration are the values that make sense for h, which are 0 to 10, so the area of the triangle is

(b) With the integral we get

With the area formula we get

What do ya know. Our methods are in agreement.

Answer

(a) Now the variable h measures the distance from the left end of the triangle to the slice. When h = 0 the slice is at the left end, and when h = 12 the slice is at the right tip.

Once again, we'll let y be the height of the slice at position h. To find y we can use similar triangles just like we did in the previous two problems. However, the base of the smaller triangle is not h. The base of the smaller triangle is now 12 – h. Using similar triangles, we have

so

The area of the slice at position h is

and the area of the triangle is

(b) Using the integral we get

Using the formula we get

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'll 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

Answer

Don't panic at the instructions "derive the area formula." All this means is we need to use integration to find the area of the triangle, and we should get

when we evaluate the integral at the end.

This problem is the same as problem 2 above except that we have letters instead of numbers for the side lengths.

Let y be the height of the slice at position x:

We have similar triangles again. The big triangle has height h and base b; the smaller triangle has height y and base x:

By similarity, we know that

Since x is the variable of integration, we need to get the height (y) of the slice in terms of x:

The area of the slice at position x is

The variable x goes from 0 at the right tip of the triangle to b at the left edge of the triangle.

Taking the integral, we find that the area of the triangle is

Let's evaluate this integral and see what we get:

We found the area of the triangle using an integral, evaluated the integral, and got the formula for the area of a triangle. Finally, the long-awaited derivation of for the area of a triangle is ours. We have the power, so we might as well use it.