§4.4 The Fundamental Theorem of Calculus

Introduction

The Fundamental Theorem of Calculus

Basic antiderivatives

The total change theorem

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.

A tank leaks at $r(t)$ gallons per day. The quantity $\int_4^{10} r(t)\,dt$ and its companion $\frac{1}{6}\int_4^{10} r(t)\,dt$ measure

A workout burns calories at a continuous rate $c(t)$ with average rate $\frac{85}{6}$ calories per minute on $[0,40]$. Solving $c(t) = \frac{85}{6}$ finds times near $5.918$ and $34.082$ minutes, guaranteed to exist because

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.

An object moves with $v(t) = 3t^2 + 40$ feet per second. Compute the exact distance traveled on $[1,5]$, using antiderivative $t^3 + 40t$.

Using the antiderivative $2x - x^2$, evaluate $\int_{-1}^{4}(2 - 2x)\,dx$.

Evaluate $\int_0^2 (3x^3 - 2x^2 - e^x)\,dx$ with antiderivative $\frac{3}{4}x^4 - \frac{2}{3}x^3 - e^x$, rounded to three decimals.

Evaluate $\int_0^1 (x^3 - x - e^x + 2)\,dx$ with antiderivative $\frac{1}{4}x^4 - \frac{1}{2}x^2 - e^x + 2x$, rounded to three decimals.

Evaluate $\int_0^{\pi/3} (2\sin t - 4\cos t + \sec^2 t - \pi)\,dt$ with antiderivative $-2\cos t - 4\sin t + \tan t - \pi t$, rounded to three decimals.

With burn rate $c(t) = -0.05t^2 + t + 10$ calories per minute on $[0,10]$, compute the total calories burned, rounded to three decimals.

The full workout's integral totals $\frac{1700}{3}$ calories over $40$ minutes. Compute the average burn rate, rounded to three decimals.

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 $[0,40]$.