Calculate \int_4^{10} 6x^2 \, dx given \int_4^9 x^2 \, dx = \frac{665}{3}, \quad \int_9^{10}...


Calculate {eq}\int_4^{10} 6x^2 \, dx {/eq} given {eq}\int_4^9 x^2 \, dx = \frac{665}{3}, \quad \int_9^{10} x^2 \, dx = \frac{271}{3} {/eq}

Interval Addition Property:

If there are two definite integrals and one of the boundaries (Upper or lower) is the same, then we can combine the integral by addition property.

  • {eq}\int_{a}^{b}f(x)\ dx+\int_{b}^{c}f(x)\ dx=\int_{a}^{c}f(x)\ dx {/eq}

Take out the constant and apply this property for this particular problem and substitute the given values to get the result.

Answer and Explanation:

Given data:

{eq}\displaystyle \int_4^9 x^2 \, dx = \frac{665}{3}\\ \displaystyle \int_9^{10} x^2 \, dx = \frac{271}{3}\\ \displaystyle \int_4^{10} 6x^2 \, dx=?\\ {/eq}

By the adding property of intervals of definite integral, we have:

{eq}\begin{align*} \int_4^{10} 6x^2 \, dx&=6\int_4^{10} x^2 \, dx\\ &=6\left (\int_4^{9} x^2 \, dx+\int_9^{10} x^2 \, dx \right ) \end{align*} {/eq}

Substitute the given required values in the above expression and simplify.

{eq}\begin{align*} 6\left (\int_4^{9} x^2 \, dx+\int_9^{10} x^2 \, dx \right )&=6\left (\frac{665}{3}+\frac{271}{3} \right )\\ &=6\left (\frac{665+271}{3}\right )\\ &=6\left (\frac{936}{3}\right )\\ &=6\left (312\right )\\ &=1872 \end{align*} {/eq}

Learn more about this topic:

Evaluating Definite Integrals Using the Fundamental Theorem

from AP Calculus AB: Exam Prep

Chapter 16 / Lesson 2

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