#### To determine

**To evaluate:**

The integral

∫-55x-25-x2dx

by interpreting it in terms of areas.

#### Answer

-25π2

#### Explanation

**1) Concept:**

A definite integral can be interpreted as a net area, that is, a difference of areas;

∫abf(x)dx=A1-A2

where A1 is the region above x- axis and below the graph of f, and A2 is the region below x- axis and above the graph of f.

**2) The property of integral**

∫abfx-gxdx=∫abf(x)dx-∫abgx dx

**3) Formula:**

Area of semi-circle = A=12(πr2)

**4) Given:**

∫-55x-25-x2dx

**5) Calculation:**

By using property of integral,

∫-55x-25-x2dx=∫-55xdx-∫-5525-x2 dx

Now, draw the graph for each part separately.

First, drawgraph of fx= x in the given domain.

The graph of y=x is the line with slope 1 as shown below.

From the above graph, the areas A1 and A2 are symmetric about the origin since the shaded area above the x axis is the same as the shaded area below the x axis. So, the net area will be zero.

Thus,

∫-55xdx=0

Now, draw graph of f(x)= 25-x2 in the given domain.

The graph of y=25-x2 in the given domain is as shown below which is a semi-circle with radius 5.

From the above graph,

A3=12πr2= 12*π*52=25π2

Since diameter is 10 and the radius is half of diameter, the radius is 5.

Therefore,

∫-55x-25-x2dx=∫-55xdx-∫-5525-x2 dx

=0-25π2=-25π2

**Conclusion:**

∫-55x-25-x2dx=-25π2