# Using area to evaluate the integral

#### Murphie

Use an area to evaluate

a^2
∫ 3x dx,
a

where a > 1

Having some trouble with this, could someone explain to me how to do this?
Thanks.

#### Sudharaka

Use an area to evaluate

a^2
∫ 3x dx,
a

where a > 1

Having some trouble with this, could someone explain to me how to do this?
Thanks.
Dear Murphie,

First draw the graph of y=3x. Since $$\displaystyle \int_{a}^{a^2}3xdx$$ represents the area of this graph from $$\displaystyle x=a$$ to$$\displaystyle x=a^2$$ you can get the area of the trapezium formed by the x-axis and the curve at the given region.

Hope this will give you an idea to solve the problem.

• mr fantastic, HallsofIvy and Murphie

#### Murphie

is the area 1/2(a)a^2 = (a^3)/2?

#### HallsofIvy

MHF Helper
No, it is not. Go back and do what Sudharaka said!

What are the coordinates of the vertices of the trapezium (I would say "trapezoid")? What is the "height" of that trapezoid? What are the lengths of the two "bases" of the trapezoid? Do you know the formula for the area of a trapezoid?

#### mr fantastic

MHF Hall of Fame
is the area 1/2(a)a^2 = (a^3)/2?
Check this answer by doing the integration.