Topic 6 of 7

Antiderivatives and the definite integral

Step 15 · Calculus 1 topics

Undo differentiation to find antiderivatives, approximate areas with Riemann sums, and read the definite integral as signed area.

Antiderivatives

F is an antiderivative of f when F′ = f. Constants differentiate to 0, so any antiderivative can be shifted up or down by a constant. We write the whole family with + C.

f(x)	Antiderivative
$x^{n},\ n \ne -1$	$\frac{x^{n+1}}{n+1} + C$
$\frac{1}{x}$	$\ln\|x\| + C$
$e^{x}$	$e^{x} + C$
$\cos x$	$\sin x + C$
$\sin x$	$-\cos x + C$

Example: A polynomial

Find all antiderivatives of 6x² − 4x + 5.

Reverse the power rule term by term: 2x³ − 2x² + 5x.
Add the constant.

Answer2x³ − 2x² + 5x + C

Example: A starting value

f′(x) = 3x² and f(1) = 4. Find f.

f(x) = x³ + C.
f(1) is 1 + C, which must be 4, so C = 3.

Answerf(x) = x³ + 3

Riemann sums

To estimate the area under y = f(x) from a to b, cut [a, b] into n strips of width Δx = (b − a) ÷ n, build a rectangle on each, and add their areas. The height can come from the left end, the right end or the middle of each strip.

\sum_{i=1}^{n} f(x_i)\,\Delta x

Four right-hand rectangles under y = x² from 0 to 2 have area 3.75. The true area is 8/3, about 2.67.

Example: A right-hand sum

Estimate the area under y = x² from 0 to 2 with 4 right-hand rectangles.

Δx = 0.5, and the right ends are 0.5, 1, 1.5 and 2.
The heights 0.25, 1, 2.25 and 4 add to 7.5.
Area ≈ 0.5 × 7.5 = 3.75.

Answer3.75, too high because x² is rising

The definite integral

As n grows, the Riemann sums close in on a single number: the definite integral. Area above the x-axis counts as positive and area below it counts as negative.

\int_{a}^{b} f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i)\,\Delta x

Example: Area from a shape

Find $\int_{0}^{4} (2x + 1)\,dx$ using geometry.

Under the line from 0 to 4 is a trapezoid with parallel sides 1 and 9, and width 4.
Its area is 4 times the average of 1 and 9, which is 4 × 5.

Answer20

Example: Signed area

Find $\int_{-2}^{2} x^{3}\,dx$ .

x³ is odd: the area below the axis from −2 to 0 matches the area above from 0 to 2.
They cancel.

Answer0

Try it yourself

Work each one on paper first, then open the answer to check.

Find all antiderivatives of 8x³.
Show answer
2x⁴ + C
Find all antiderivatives of $\frac{1}{x} + e^{x}$ .
Show answer
$\ln|x| + e^{x} + C$
f′(x) = 4x − 1 and f(0) = 3. Find f(x).
Show answer
2x² − x + 3
Estimate the area under y = x from 0 to 4 with 4 left-hand rectangles.
Show answer
6
Find $\int_{0}^{3} 4\,dx$ .
Show answer
12
Find $\int_{0}^{6} x\,dx$ using a triangle.
Show answer
18
$\int_{0}^{5} f(x)\,dx = 7$ and $\int_{0}^{2} f(x)\,dx = 3$ . Find $\int_{2}^{5} f(x)\,dx$ .
Show answer
4