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Area Between Two Curves Calculator

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Area between two curves calculator will help to calculate the area of the region covered under the two interdected curves. Area is computed by the definite integral. If y = f(x) and y = g(x) are the two curves on the interval [a,b], such that f(x)>g(x), then the area bounded by the lines x = a and x = b is,

Area = $\int_{a}^{b}[upper \ curve - lower \ curve]dx$

Area = $\int_{a}^{b}[f(x) - g(x)]dx$

Plug in the two functions with the intervals into the area between two curves calculator and area is calculated.
 

Steps

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Step 1 : Observe the two curves f(x) and g(x) and equate the both the curve to find the intersecting points.

Step 2 : Substitute the intersecting points in the area between the two curve formula given above. Simplify by substituting the limit and get the area bounded.

Problems

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Below are the problems based on Area Between two curves.

Solved Examples

Question 1: Determine the are bounded by the curve $y = 2x^2 + 10$ and $y = 4x + 16$ .
Solution:
Step 1 : Given $f(x) = 2x^2 + 10$  
$g(x) = 4x + 16$

$2x^2 + 10 = 4x + 16$

$2x^2 - 4x - 6 = 0$

$2(x+1)(x-3) = 0$

The two intersecting points are x = -1 and x = 3.

Step 2 : Area bounded between the two curve,
Area = $\int_{a}^{b}[f(x)-g(x)]dx$

       = $\int_{-1}^{3}[(2x^2 + 10)-(4x + 16)]dx$

       = $\int_{-1}^{3}[2x^2 - 4x - 6]dx$

       = [2 $\frac{x^{3}}{3}$ - 2 ${x^{2}} - 6x]^{3}_{2}$ 

       = - $\frac{64}{3}$

Therefore, the area bounded between the two curve is - $\frac{64}{3}$.


Question 2: Determine the are bounded by the curve $y = x^2$ and $y = \sqrt{x}$ .
Solution:
Step 1 :Given $f(x) = \sqrt{x}$
$g(x) = x^2$

$\sqrt{x} = x^2$
$x = 0, 1$

The intersecting points are x = 0, 1.

Step 2 : Area bounded between the two curve,

Area = $\int_{a}^{b}[f(x)- g(x)]dx$

       = $\int_{-1}^{3}[(\sqrt{x})-(x^2)]dx$

       = [$\frac{2}{3}$$x^{\frac{3}{2}}$-$\frac{1}{3}$$x^{3}]^{1}_{0}$

       = $\frac{1}{3}$

Therefore, the area bounded between the two curve is $\frac{1}{3}$.