# Area of Function Problems

### Solutions

1 Calculate the area of the site bounded by the curve y = 4x − x^{2} and the x-axis.

2 Find the area of the plane region enclosed by the curve y = ln x between the point of intersection with the x-axis and x = e.

3Find the area bounded by the line x + y = 10, the x-axis, x = 2 and x = 8.

4Calculate the area enclosed by the curve y = 6x^{2} − 3x^{3} and the x-axis.

5 Calculate the area enclosed by the curve f(x) = x^{3} − 6x^{2} + 8x and the x-axis.

6 Calculate the area of a circle of radius r.

7Find the area of an ellipse of semiaxes a and b.

8 Calculate the area enclosed by the curve y = x^{2} − 5x + 6 and the line y = 2x.

9 Calculate the area enclosed by the parabola y^{2} = 4x and the line y = x.

10Calculate the area enclosed by 3y = x^{2} and y = −x^{2} + 4x.

11 Calculate the area enclosed by y= x^{2} − 2x and y = −x^{2} + 4x.

12Calculate the area enclosed by:

y = sin x, y = cos x, x = 0.

## 1

Calculate the area of the site bounded by the curve y = 4x − x^{2} and the x-axis.

First, find the x-intercepts to the curve and the limits of integration.

## 2

Find the area of the plane region enclosed by the curve y = ln x between the point of intersection with the x-axis and x = e.

First, find the x-intercepts.

## 3

Find the area bounded by the line x + y = 10, the x-axis, x = 2 and x = 8.

## 4

Calculate the area enclosed by the curve y = 6x^{2} − 3x^{3} and the x-axis.

## 5

Calculate the area enclosed by the curve f(x) = x^{3} − 6x^{2} + 8x and the x-axis.

The area, for reasons of symmetry, can be written as:

## 6

Calculate the area of a circle of radius r.

Start from the equation of the circle x² + y² = r².

The area of the circle is four times the area of the first quadrant.

Calculate the indefinite integral by change of variable.

Find the new limits of integration.

## 7

Find the area of an ellipse of semiaxes a and b.

As the ellipse is a symmetrical curve, the area requested will be 4 times the area enclosed in the first quadrant of the coordinate axes.

Find the new limits of integration.

## 8

Calculate the area enclosed by the curve y = x^{2 } −5x + 6 and the line y = 2x.

First, find the points of intersection of the two functions to know the limits of integration.

From x = 1 to x = 6, the line is above the parabola.

## 9

Calculate the area enclosed by the parabola y^{2} = 4x and the line y = x.

From x = 0 to x = 4, the parabola is above the line.

## 10

Calculate the area enclosed by 3y = x^{2} and y = −x^{2} + 4x.

First, represent the parabolas from the vertex and the points of intersection with the axes.

Also, find the points of intersection of the functions, which will give the limits of integration.

## 11

Calculate the area enclosed by y= x^{2} − 2x and y = −x^{2} + 4x.

Represent the parabolas from the vertex and the points of intersection with the axes.

## 12

Calculate the area enclosed by:

y = sin x, y = cos x, x = 0.

First, find the points of intersection of the functions:

The cosine graph is above the graph within the limits of integration.