5.3: Fundamental Theorem of Calculus

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Throughout this module, if something does not exist, write DNE in the answer box.

Recap Video

Take a look at the following video which recaps the ideas from the section.

Example Video

Below are videos showing two worked examples.

Problems

Let $f(x)$ be a continuous function on an interval $[a,b]$ .

Part 1: If $\displaystyle F(x) = \int _a^x f(t)dt$ . Then $F(x)$ is continuous on $[a,b]$ , differentiable on $(a,b)$ , and $F'(x) = f(x)$ .
Part 2: If $F(x)$ is any antiderivative of $f(x)$ , then $\displaystyle \int _a^b f(x)dx = F(b)-F(a)$ .

Let $F(x)$ be the function given by $\displaystyle F(x) = \int _{-1}^x \sqrt {1+t^4}\,dt$ . Then $F'(x) = \answer {\sqrt {1+x^4}}$ .

Let $g(x)$ be the function given by $\displaystyle g(x) = \int _0^{\sqrt {x}} \sin (t^2)\,dt$ . Then $g'(x) = \answer {\frac {\sin (x)}{2\sqrt {x}}}.$

You need to use the chain rule.

Let $h(x)$ be the function $\displaystyle h(x) = \int _{x^2}^{x^4} e^{\sqrt {t}}\,dt$ . Then $h'(x) = \answer {4x^3e^{x^2}-2xe^x}$ .

Consider a function $f(x)$ whose graph is shown below.

Let $\displaystyle g(x) = \int _{-1}^x f(t)dt$ for $x > -1$ .

What are the critical numbers of $g(x)$ ? List them in increasing order: $x = \answer {0} \quad \text { and } \quad x = \answer {3}.$
On what interval(s) is $g(x)$ increasing?
$(0,3)$ only $(-1,0)$ only $(-1,0)$ and $(3,\infty )$ $(2,\infty )$ only $(3,\infty )$ only.
On what interval(s) is $g(x)$ decreasing?
$(0,3)$ only $(-1,0)$ and $(0,3)$ $(-1,0)$ and $(3,\infty )$ $(2,\infty )$ only $(3,\infty )$ only.
There is a local minimum of $g(x)$ at $x = \answer {3}$ .
On what interval(s) is $g(x)$ concave up?
$(0,2)$ only $(-1,0)$ only $(-1,0)$ and $(2,\infty )$ $(2,\infty )$ only $(3,\infty )$ only.
On what interval(s) is $g(x)$ concave down?
$(0,2)$ only $(-1,0)$ only $(-1,0)$ and $(2,\infty )$ $(2,\infty )$ only $(3,\infty )$ only.
There are inflection points of $g(x)$ at (in increasing order) $x = \answer {0}$ and $x = \answer {2}$ .

Using FTC Part 2, evaluate $\displaystyle \int _0^3 x^2\,dx$ .

First, we find an antiderivative of $f(x) = x^2$ . Which of the following represents an antiderivative $F(x)$ of $f(x) = x^2$ ? Select all that apply.

$F(x) = x^3/3$ $F(x) = 2x$ $F(x) = 2x+1$ $F(x) = x^3/3-1$

Notice that it doesn’t matter which one we take, so let’s use $F(x) = \frac {x^3}{3}$ . We now plug in the endpoints into $F(x)$ and subtract: $\int _0^3 x^2\,dx = \frac {x^3}{3}\Big |_0^3 = \answer {9}.$

The integral $\displaystyle \int _0^{\pi } \sin (x)\,dx = \answer {2}$ .