During a thunderstorm, rain was falling at the rate of 8 √(2 t)-32 t^3 (0 ≤t ≤0.6) in./hr. a.

step 1 All right, so what they tell us is the rate of rainfalling. So whenever it's a rate, it's a deri

During a thunderstorm, rain was falling at the rate of 8 √(2...

During a thunderstorm, rain was falling at the rate of $$8 \sqrt{2 t}-32 t^{3} \quad(0 \leq t \leq 0.6)$$ in./hr. a. Find an expression giving the total amount of rainfall after $t$ hr. Hint: The total amount of rainfall at $t=0$ is zero. b. How much rain had fallen after $\frac{1}{2} \mathrm{hr}$ ?

During a thunderstorm, rain was falling at the rate of
$$8 \sqrt{2 t}-32 t^{3} \quad(0 \leq t \leq 0.6)$$
in./hr.
a. Find an expression giving the total amount of rainfall after $t$ hr.
Hint: The total amount of rainfall at $t=0$ is zero.
b. How much rain had fallen after $\frac{1}{2} \mathrm{hr}$ ?

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Key Concepts

Integration

Integration is the mathematical process of finding the accumulation of a quantity, which, in this context, means summing up the instantaneous rate of rainfall over time to obtain the total amount of rainfall. It converts a rate function into a cumulative quantity by calculating the area under the curve of the rate function.

Rate Function

A rate function describes how quickly a quantity is changing—in this case, the rainfall rate in inches per hour. Understanding the rate function is essential because it provides the information that needs to be integrated over time to determine the total accumulation of rainfall.

Definite Integral

A definite integral is used to compute the accumulation of a quantity over a specific interval. By integrating the rate function from the starting time to a given time, one finds the total amount of rainfall accumulated during that time period. This concept is key to relating the instantaneous rate to the overall effect.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation with integration. It states that if a function is integrated over an interval and then differentiated, one recovers the original function (the rate function, in this case). This theorem justifies using integration to determine the net accumulation of rainfall from a given rate.

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