§4.4 The Fundamental Theorem of Calculus

Introduction

The Fundamental Theorem of Calculus

Basic antiderivatives

The total change theorem

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

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 $4 \le t \le 10$?

Evaluate $\int_0^{\frac{\pi}{2}}\sin(x)\,dx$.

Using the fact that the exponential is its own antiderivative, evaluate $\int_0^1 e^x\,dx$.

Use symmetry of the fifth power to evaluate $\int_{-1}^{1}x^5\,dx$ without any computation.

Evaluate $\int_0^1(\sqrt{x} - x^2)\,dx$ as a fraction.

A balloon tossed from $32$ feet with $v(t) = -32t + 16$ has position $s(t) = -16t^2 + 16t + 32$. Compute its maximum height in feet.

If $C$ is an antiderivative of a calorie burn rate $c(t)$, then $C(40) - C(0) = \int_0^{40}c(t)\,dt$ is the total calories burned on $[0,40]$.