How do you find the area under the graph of $f (x) = x^{2}$ $f(x)=x^2$ on the interval $[- 3, 3]$ $[-3,3]$ ?

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How do you find the area under the graph of $f (x) = x^{2}$ $f(x)=x^2$ on the interval $[- 3, 3]$ $[-3,3]$ ?

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Answer 1 · 2014-09-17T22:51:25

This is an integration problem. We will find the area under the curve $x^{2}$ $x^2$ over the interval [-3,3]. This included both -3 and 3 because of the square brackets.

The integration of $x^{2}$ $x^2$ is found by incrementing the power to $3$ $3$ and using $3$ $3$ as the denominator.

$\int_{a}^{b} x^{n} d x = {[\frac{x^{n + 1}}{n + 1}]}_{a}^{b} = \frac{b^{n + 1}}{n + 1} - \frac{a^{n + 1}}{n + 1}$ $int_a^bx^ndx=[x^(n+1)/(n+1)]_a^b=b^(n+1)/(n+1)-a^(n+1)/(n+1)$

$\int_{- 3}^{3} x^{2} d x = {[\frac{x^{3}}{3}]}_{- 3}^{3} = [\frac{{(3)}^{3}}{3} - \frac{{(- 3)}^{3}}{3}] = \frac{27}{3} - \frac{- 27}{3} = \frac{27}{3} + \frac{27}{3} = 9 + 9 = 18$ $int_-3^3x^2 dx=[x^3/3]_-3^3=[(3)^3/3-(-3)^3/3]=27/3-(-27)/3=27/3+27/3=9+9=18$

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How do you find the area under the graph of $f (x) = x^{2}$ $f(x)=x^2$ on the interval $[- 3, 3]$ $[-3,3]$ ?

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