Integration: Area under a curve

[ area diff. axes ][ posit. & neg. area ][ between curves]





Area under a curve related to different axes

curve area

curve are equation#1

Example #1

Find the area 'A' enclosed by the x-axis, x=2, x=4 and the graph of y=x3/10.

curve area problem #1

curve area problem#1 - solution

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Positive and negative area

curve area #2

curve area - negative area

note: This expression calculates the absolute area between the curve the vertical lines at 'a' and 'b' and the x-axis. It takes no account of sign. If sign were an issue then the two integrals on the first line would be added and not subtracted.

Unless told differently, assume that the absolute area is required.

Example #1

Find the area 'A' enclosed by the x-axis, x=1, x=8 and the graph of y=2sin[(x+3)/2].

area under curve problem#2

The curve crosses the x-axis at y=0.

Therefore 2sin[(x+3)/2]=0

Sine is zero when the angle is 0,180 or 360 deg.

(zero, pi and 2 pi)

area unser curve problem 2e

area under curves problem#2d

area under curves problem#2f


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Area bounded by two curves

curve area #3

curve area - included between curves

Example #1

To 3 d.p. calculate the area 'A' included between the curves y=x2/2 and y=(0.75)x

first find the x value where the curves cross

included area problem#3a

included area problem


The area 'A' is the difference between the area under the straight line and the area under the parabola, from x=0 to x=3.5 .

included area problem 3b


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