Area of Function Problems

Solutions

1 Calculate the area of the site bounded by the curve y = 4x − x2 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 = 6x2 − 3x3 and the x-axis.

5 Calculate the area enclosed by the curve f(x) = x3 − 6x2 + 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 = x2 − 5x + 6 and the line y = 2x.

9  Calculate the area enclosed by the parabola y2 = 4x and the line y = x.

10Calculate the area enclosed by 3y = x2 and y = −x2 + 4x.

11 Calculate the area enclosed by y= x2 − 2x and y = −x2 + 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 − x2 and the x-axis.

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

Area of Function Operations

Area of Function Problem

Area of Function Solution


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.

Area of Function Problem

First, find the x-intercepts.

Area of Function Operations

Area of Function Operations

Derive

Integrate

Area of Function Operations

Area of Function Solution


3

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

Area of Function Problem

Area of Function Solution


4

Calculate the area enclosed by the curve y = 6x2 − 3x3 and the x-axis.

Area of Function Operations

Area of Function Problem

Area of Function Solution


5

Calculate the area enclosed by the curve f(x) = x3 − 6x2 + 8x and the x-axis.

Area of Function Operations

Area of Function Operations

Area of Function Problem

Area of Function Operations

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

Area of Function Solution


6

Calculate the area of a circle of radius r.

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

Area of Function Problem

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

Area of Function Operations

Calculate the indefinite integral by change of variable.

Area of Function Operations

Change of Variable

Area of Function Operations

Area of Function Operations

Area of Function Operations

Find the new limits of integration.

Area of Function Operations

Area of Function Operations

Area of Function Operations

Area of Function Solution


7

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

Area of Function Problem

Area of Function Operations

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.

Area of Function Operations

Area of Function Operations

Change of Variable

Area of Function Operations

Area of Function Operations

Area of Function Operations

Find the new limits of integration.

Area of Function Operations

Area of Function Operations

Area of Function Solution


8

Calculate the area enclosed by the curve y = x2 −5x + 6 and the line y = 2x.

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

Area of Function Operations

Area of Function Problem

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

Area of Function Operations

Area of Function Solution


9

Calculate the area enclosed by the parabola y2 = 4x and the line y = x.

Area of Function Operations

Area of Function Problem

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

Area of Function Solution


10

Calculate the area enclosed by 3y = x2 and y = −x2 + 4x.

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

Area of Function Operations

Area of Function Operations

Area of Function Operations

Area of Function Operations

Area of Function Operations

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

System of Equations

Area of Function Problem

Area of Function Operations

Area of Function Solution


11

Calculate the area enclosed by y= x2 − 2x and y = −x2 + 4x.

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

Area of Function Operations

Area of Function Operations

Area of Function Operations

Area of Function Operations

Area of Function Operations

Area of Function Problem

Area of Function Operations

Area of Function Operations

Area of Function Operations

Area of Function Solution


12

Calculate the area enclosed by:

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

First, find the points of intersection of the functions:

Points of Intersection

Area of Function Problem

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

Area of Function Solution