H2 Maths 5.4 - Area Between Curves

Step 1 - Set up the problem

We want the area enclosed between y equals x squared and y equals two x plus three.

Step 1 - Set up the problem

Both are familiar shapes: a parabola opening upward, and a straight line.

Step 1 - Set up the problem

Before integrating, we need to know where they cross.

Step 2 - Find the intersection points

Set the two expressions equal and rearrange.

Step 2 - Find the intersection points

X squared minus two x minus three equals zero.

Step 2 - Find the intersection points

Factor as x minus three times x plus one, giving x equals three and x equals negative one.

Step 2 - Find the intersection points

These are the left and right boundaries of the enclosed region.

Step 3 - Identify the top curve and set up the integral

Between x equals negative one and x equals three, we need to know which curve sits on top.

Step 3 - Identify the top curve and set up the integral

Test x equals zero: the line gives three, the parabola gives zero.

Step 3 - Identify the top curve and set up the integral

So the line is above the parabola throughout the region.

Step 3 - Identify the top curve and set up the integral

The area is the integral of top minus bottom, from negative one to three.

Step 4 - Evaluate the integral

Expand and integrate term by term.

Step 4 - Evaluate the integral

The antiderivative of two x plus three minus x squared is x squared plus three x minus x cubed over three.

Step 4 - Evaluate the integral

Substitute the upper limit three: nine plus nine minus nine gives nine.

Step 4 - Evaluate the integral

Substitute the lower limit negative one: one minus three plus one third gives negative five thirds.

Step 4 - Evaluate the integral

Subtract lower from upper: nine minus negative five thirds equals thirty-two thirds.

Step 5 - State the final answer and exam pitfall

The enclosed area is 32 over 3 square units.

Step 5 - State the final answer and exam pitfall

Common pitfall: always check which curve is on top before integrating.

Step 5 - State the final answer and exam pitfall

If you integrate x squared minus two x plus three instead, you get the negative of the correct answer.

Step 5 - State the final answer and exam pitfall

Sketch the region, label the intersection points, and confirm the sign before you write down the integral.