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New Zealand
Level 8 - NCEA Level 3

Area bound by curves and x/y axis, area between two curves (sin/cos)


Using what we know about finding the area bounded by a curve and the x-axis, and also the area bounded by two curves, we'll now specifically look at calculating areas where the functions involved are sine and cosine.

We'll be using our exact values for sine and cosine so you might like to become reacquainted with this all important table:

  $0$0 $\frac{\pi}{6}$π6 $\frac{\pi}{4}$π4 $\frac{\pi}{3}$π3 $\frac{\pi}{2}$π2
$\sin$sin $0$0 $\frac{1}{2}$12 $\frac{1}{\sqrt{2}}$12 $\frac{\sqrt{3}}{2}$32 $1$1
$\cos$cos $1$1 $\frac{\sqrt{3}}{2}$32 $\frac{1}{\sqrt{2}}$12 $\frac{1}{2}$12 $0$0

The area bounded by the curve and the x-axis

Due to the beautifully symmetric nature of our sine and cosine curves, we can often find easy ways to calculate bounded area.

Example 1

Consider the function $y=3\cos2x$y=3cos2x

Find the area bounded by the curve and the x-axis over the interval $0<=x<=\frac{3\pi}{4}$0<=x<=3π4

We could break up our area into the region above and the region below the $x$x-axis and combine the two together. But far easier is to realise that the total area is three times the area from $0<=x<=\frac{\pi}{4}$0<=x<=π4


The area bounded between two curves

Worked Examples

Example 1

Consider the area bounded by the graphs $y=\sin2x$y=sin2x and $y=\cos x$y=cosx on the graph below.

To find the shaded area we could again break it into two parts, or instead we can realise that both sections contain the same area, leaving us with a more simple calculation.

Remember, since we have two curves, we need to subtract the area under the lower curve from the area under the upper curve.

Putting it all together we can calculate as follows:


example 2

Consider the area bounded by the graphs $y=\cos x$y=cosx and $y=\cos2x$y=cos2x on the graph below.

To find the shaded area, we need to find the area under the top curve and subtract from that the area under the bottom curve. 

We can calculate as follows:


Worked Examples

Question 1

Find the exact area of the shaded regions bounded by the curve $y=3\cos x$y=3cosx.

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Question 2

Find the area bounded by the curve $y=3\cos x$y=3cosx and the $x$x-axis between $x=0$x=0 and $x=\frac{\pi}{2}$x=π2.

Question 3

Find the area enclosed by the curves $y=\sin x$y=sinx and $y=\cos x$y=cosx, and the lines $x=0$x=0 and $x=\frac{\pi}{4}$x=π4.

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Question 4

The two functions $y=5\cos2x$y=5cos2x and $y=5\sin\left(\frac{x}{2}\right)$y=5sin(x2) meet at $x=\frac{\pi}{5}$x=π5 and $x=\pi$x=π.

Find the area bounded by the two curves. Give your answer to two decimal places.

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