Area between Curves: Parabola and a Linear

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Find the area between Equations of parabola and line

What is the shape of

It is a parabola along the x-axis.

Does it pass through the origin?

No, it is offset 2 units to the left.

Draw a graph of

on paper. Then check your diagram by clicking “Next”.

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Add the line

to your diagram. Then check your diagram by clicking “Next”.

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Check the coordinates of the apparent points of intersection of the two curves in the equations.

For

is OK.

For (2, 2):

is OK.

What is the importance of the point

It divides the area into sections in which the description of the bottom of the area is defined differently, as shown in this diagram.
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How about the top of the area?

The top of the area is defined in the same way in both sections.

How is that?

The upper branch of the parabola defines the top of the area everywhere.

State the equation which defines the top.

How is the bottom of the area defined on the left?

It is defined by the lower branch of the parabola.

State that equation.

How is the bottom of the area defined on the right?

It is defined by the line.

State that equation.

How do we determine the area?

We integrate over these two regions using tiny rectangles as shown in this diagram.
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How wide are the rectangles?

Their width is dx

How do we determine their height?

We subtract the y-coordinate of the bottom from the y-coordinate of the top.

Set up the integral over the left (green) region in this diagram.

Insert the limits of integration.

Simplify this integral.

Integral for the area bounded by the parabola

What method can we use to evaluate this integral?

We can substitute

, and du = dx.

Do that.

We get

. (We’ll switch back to x to put in the limits.)

What method can we now use?

We can rewrite the integral as a power and use the Power Rule.

Do that.

Do the integration.

We get

Convert this back to an expression in terms of x and show the limits we must use.

The area becomes

Substitute the limits.

We get

Now let’s find the area of the right (red) section. Set up the integral over the right region in the diagram.

Insert the limits of integration.

Integral for the area bounded by the parabola and the line

How can we do this integral?

For the first term we can use our previous result (without the factor 2). We can use the power rule on the second term.

Do that.

Substitute the limits.

Are we finished?

No.

What next?

We want the total area. How do we get that?

We add the two areas just calculated.

Do that.

We get

How can we check this result?

We can count squares on the graph.

Does 4.5 square units seem reasonable?

Yes.

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