§4.4 The Fundamental Theorem of Calculus

Introduction

The Fundamental Theorem of Calculus

Basic antiderivatives

The total change theorem

A student with a third floor dormitory window 32 feet off the ground tosses a water balloon straight up in the air with an initial velocity of 16 feet per second. It turns out that the instantaneous velocity of the water balloon is given by $v(t) = -32t + 16$, where $v$ is measured in feet per second and $t$ is measured in seconds.

Let $s(t)$ represent the height of the water balloon above ground at time $t$, and note that $s$ is an antiderivative of $v$. That is, $v$ is the derivative of $s$: $s'(t) = v(t)$. Find a formula for $s(t)$ that satisfies the initial condition that the balloon is tossed from 32 feet above ground. In other words, make your formula for $s$ satisfy $s(0) = 32$.

Determine the exact distance traveled on $[1,5]$ by an object with velocity function $v(t) = 3t^2 + 40$ feet per second.

$\int_{-1}^4 (2-2x) \, dx$

$\int_{0}^{\frac{\pi}{2}} \sin(x) \, dx$

$\int_0^1 e^x \, dx$

$\int_{-1}^{1} x^5 \, dx$

$\int_0^2 (3x^3 - 2x^2 - e^x) \, dx$

$\displaystyle \int_0^1 \left(x^3 - x - e^x + 2\right) \,dx$

$\displaystyle \int_0^{\pi/3} (2\sin (t) - 4\cos(t) + \sec^2(t) - \pi) \, dt$

$\displaystyle \int_0^1 (\sqrt{x}-x^2) \, dx$

Suppose that pollutants are leaking out of an underground storage tank at a rate of $r(t)$ gallons/day, where $t$ is measured in days. It is conjectured that $r(t)$ is given by the formula $r(t) = 0.0069t^3 -0.125t^2+11.079$ over a certain 12-day period. What is the meaning of $\int_4^{10} r(t) \, dt$ and what is its value? What is the average rate at which pollutants are leaving the tank on the time interval $4 \le t \le 10$?

What is the exact total number of calories the person burns during the first 10 minutes of her workout?

Let $C(t)$ be an antiderivative of $c(t)$. What is the meaning of $C(40) - C(0)$ in the context of the person exercising? Include units on your answer.

Determine the exact average rate at which the person burned calories during the 40-minute workout. Sketch a representation of the average rate at which calories were burned on the provided graph of $c(t)$.

At what time(s), if any, is the instantaneous rate at which the person is burning calories equal to the average rate at which she burns calories, on the time interval $0 \le t \le 40$?