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Areas Under Curve

Areas Under Curve :

 In this section , there is only one curve above x-axis . We need to find the area of that curve in given interval . Let f(x) be the curve above x-axis .

The area of f(x) between x=a to x= b is \( \int_{a}^{b} f(x) dx  \) is the formula .

If f(x) is below x-axis then we get Area of f(x) =\( \int_{a}^{b} -f(x) dx \).

Let us observe some of the questions on this .

Q1) Find the area of the function f(x)=x+1 between x=0 to 3.

Sol) Area = \( \int_{0}^{3} f(x) dx\)

              =\( \int_{0}^{3} (x+1)dx \)

               =\( [\frac{x^{2}}{2} + x ]_{0}^{3} \)

               = \( \frac{3^2}{2} + 3 - 0\)

               = \( \frac{9}{2} + 3 \)

               =\( \frac{15}{2} \).

 Therefore the area of the function f(x) is 15/2 =7.5 sq. units.

Q2) Find the area of the curve f(x)=4-x^2 between x=-2 to x=2 .

 Sol) Area = \( \int_{-2}^{2} f(x) dx \)

               = \( \int_{-2}^{2} (4-x^{2}) dx \)

               = \( [4x - \frac{x^{3}}{3}]_{-2}^{2} \)

                =\( 4(2)-\frac{(2)^{3}}{3} - 4(-2) + \frac{(-2)^{3}}{3} \)

               = \( 8 - \frac{8}{3} + 8 - \frac{8}{3} \)

               =\( 16 - \frac{16}{3}\)

               =\( \frac{32}{3}\) sq.units.

Q3) Find the area of the curve f(x)=4-x between x=-1 to  x=2.

Sol) Area = \( \int_{-1}^{2}f(x) dx \)

              = \( \int_{-1}^{2} 4-x dx \)

              = \( [ 4x - \frac{x^{2}}{2}]_{-1}^{2} \)

              = \( 4(2) - \frac{2^{2}}{2} - 4(-1) + \frac{(-1)^{2}}{2} \)

               = \( 8 - 2 + 4 + \frac{1}{2} \)

              = \( \frac{21}{2} \) sq. units.

Q4) Find the area of the curve f(x) = Cos x   between x=\( \frac{-\prod}{2} \) to x= \( \frac{\prod}{2}\).

Solution: 

    Area = \( \int_{\frac{-\prod}{2}}^{\frac{\prod}{2}} cos x dx \)

            = \( [Sin x ]_{\frac{-\prod}{2}}^{\frac{\prod}{2}} \)

            = \( Sin( \frac{\prod}{2}) - Sin ( \frac{-\prod}{2} ) \)

            = 1 - ( -1)

            = 2

Areas Under Curve



Sample Problems Related To Areas

Click the link Below to View Them

1)http://www.mathmaa.com/support-files/areasusingintegrals.pdf



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