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Question

# The area of an oil spill is increasing at a rate of 25t$\mathrm { ft } ^ { 2 } / \mathrm { s }$seconds after the spill. Between times t=2 and t=4 the area of the spill increases by ___$ft^2$.

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$\color{#4257b2} 25 t \;\;$ is the rate at which the area of the spill is changing at time t

The area of the spill increases by $\color{#4257b2} \int_{2}^{4} 25 t \,dx$

\begin{align*} \int_{2}^{4} 25 t \,dx&=25 \int_{2}^{4} t \,dx\\\\ &=25\dfrac{t^2}{2}\bigg|_{2}^{4}\\\\ &=25\left(\dfrac{4^2-2^2}{2} \right)\\\\ &=25\cdot \dfrac{12}{2}\\\\ &=\color{#4257b2} 150 \end{align*}

Therefore, The area of the spill increases by$\; \color{#4257b2}\boxed{ 150 ft^2}$

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