Area of a Region Bounded by Curves

Solved Problems

Solution.

It is easy to see that the curve $$ and the straight line $$ intersect at the points $$ $$ (Figure $$).

Then

Solution.

First we find the points of intersection of both curves:

Now we calculate the area $$ using integration:

Solution.

The upper graph is $$ and the lower curve is $$ Hence, the area is

Solution.

Since each petal has the same area, we calculate the area of one petal and multiply the result by three. So we have

Hence, the area of the all region is $$

Solution.

We can easily the area of the cardioid by integrating the polar equation in the interval $$ This yields:

Solution.

We represent the equation of the astroid in parametric form:

Check by substitution:

which is true.

Let's now calculate the area of the region enclosed by the parametric curve.

We apply the following integration formula:

As

we have