Integration Finding the Area Under a Curve & the Area Between Two Lines AS Maths with Liz

Note to self – begin ppt with definite integrals algebraically next year!!!!

In previous lessons… We learned how to evaluate integrals between certain values known as upper and lower bounds. Start with this one as a review Evaluate

We will be using definite integration to work out the area: In today’s lesson… We will be using definite integration to work out the area: under a curve between a line and a curve between two curves

Example 1 Find the area between the curve and the x-axis between the values 0 and 3. Solution: Method 1: Estimation We can split the figure into familiar shapes such as rectangles and add up the areas of each. BUT, this is an under-estimate! BUT, this is an over-estimate!

How can we get an accurate area? Method 2: Integration

You try Find the area under the curve between and .

Note If you’re asked to work out the area and you end up with a negative area, you must take the absolute value as your final answer (the positive version) because there is no such thing as a negative area!

Example 2 Find the area enclosed by the curve and the line .

Example 3 Find the area enclosed by the curve and the line .

You try! Find the points of intersection and the shaded areas. (a)

You try! Find the points of intersection and the shaded areas. (b)

Example 4 – Extension (Core 2) Find the area bounded by the curves and .

Example 5 – Extension (Core 2) Find the area bounded by the curves and .