How do you find the limits for the area between a line and a curve?

Equate the two equations and solve for the x-coordinates of the intersections.

How do you decide which graph is upper?

Sketch, or test an x-value between the limits in both equations.

What does a negative answer for an area between graphs suggest?

You have subtracted the wrong way round (lower minus upper).

Area between y=6x-x^2 and y=2x?

\int0^4(4x-x^2)\,dx=\frac{32}{3}.

Does the region have to lie above the x-axis for upper minus lower to work?

No; the difference of heights is positive wherever one graph is above the other.

Alternative way to find the area between a line and a curve?

Area under the upper graph minus area under the lower graph, between the same limits.

When can geometry replace integration?

For the area under a straight line, which forms a trapezium or triangle.

Area between a curve and a line or two curvesEdexcel International A Level Maths: Flashcards

Card 1 of 120 of 12 known

Question

Area between an upper graph and a lower graph?

Tap or press Space to reveal

Tap card or press Space to flip

See all 12 cards

Area between an upper graph and a lower graph?: $\int_a^b[\text{upper}-\text{lower}]\,dx$ .
How do you find the limits for the area between a line and a curve?: Equate the two equations and solve for the $x$ -coordinates of the intersections.
How do you decide which graph is upper?: Sketch, or test an $x$ -value between the limits in both equations.
What does a negative answer for an area between graphs suggest?: You have subtracted the wrong way round (lower minus upper).
Intersections of $y=6x-x^2$ and $y=2x$ ?: $x=0$ and $x=4$ .
Area between $y=6x-x^2$ and $y=2x$ ?: $\int_0^4(4x-x^2)\,dx=\frac{32}{3}$ .
Intersections of $y=x^2$ and $y=x+2$ ?: $x=-1$ and $x=2$ .
Area between $y=x^2$ and $y=x+2$ ?: $\frac92$ .
Does the region have to lie above the $x$ -axis for upper minus lower to work?: No; the difference of heights is positive wherever one graph is above the other.
Intersections of $y=x^2$ and $y=8-x^2$ ?: $x=\pm2$ , at $(\pm2,4)$ .
Alternative way to find the area between a line and a curve?: Area under the upper graph minus area under the lower graph, between the same limits.
When can geometry replace integration?: For the area under a straight line, which forms a trapezium or triangle.

Exam questions on Area between a curve and a line or two curves

The line $y=2x$ meets the curve $y=6x-x^2$ at the origin $O$ and at the point $A$ .
Find the area of the finite region bounded by the line and the curve.2 marks
The curve $y=x^2$ and the line $y=x+2$ intersect at the points $P$ and $Q$ .
Find the area of the region bounded by the curve $y=x^2$ , the $x$ -axis and the lines $x=-1$ and $x=2$ .2 marks
The curve $C$ has equation $y=x^2-4x+5$ and the line $l$ has equation $y=x+1$ .
Find the coordinates of the points where $l$ meets $C$ .3 marks

See the full worksheet

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).