Jupiter2U School
Chapter 16

16. Integration

College

Integration is differentiation run backwards, and simultaneously a way to add up infinitely many infinitesimal pieces. It computes areas, totals and accumulations — completing the two halves of calculus.

At a glance
Core ideaIntegration adds infinitely many thin slices — and undoes differentiation.
Key termFundamental Theorem: ∫ₐᵇ f = F(b) − F(a) whenever F′ = f.
You can…Find areas, totals and accumulations via antiderivatives and definite integrals.
Watch outNever forget the + C on an indefinite integral.
01 · Theory

Antiderivatives and the definite integral

An antiderivative of f is a function F whose derivative is f. The indefinite integral collects all of them (they differ by a constant):

∫ xⁿ dx = x^(n+1)/(n + 1) + C   (n ≠ −1)
∫ (1/x) dx = ln|x| + C

The definite integral ∫ab f(x) dx is the signed area between the curve and the x-axis from a to b, defined as the limit of summing thin rectangles (a Riemann sum). The two ideas are joined by the crowning result of calculus:

a b
A Riemann sum approximates the area under a curve with thin rectangles — shrinking their width toward zero gives the exact definite integral.
Fundamental Theorem of CalculusIf F′ = f, then ∫ab f(x) dx = F(b) − F(a). Integration and differentiation are inverse operations.
02 · Explanation

Adding up infinitely thin slices

To find the area under a curve, chop the region into many thin vertical strips, each almost a rectangle of height f(x) and width dx. Add their areas and you approximate the total; let the strips become infinitely thin and the approximation becomes exact. That limiting sum is the integral — the elongated S of ∫ is literally an old-style "S" for "sum."

The astonishing part is the Fundamental Theorem: this messy infinite sum can be computed by finding an antiderivative and subtracting two values. It says accumulation (integration) and rate-of-change (differentiation) are two sides of one coin. Intuitively, if you know your speed at every instant (a rate), integrating it gives the total distance travelled (an accumulation) — and that is exactly F(b) − F(a).

03 · Practical

Worked example: area under a parabola

Problem

Find the area under y = x² from x = 0 to x = 3.

  1. Set up the definite integral: ∫03 x² dx.
  2. Find an antiderivative: F(x) = x³/3 (since d/dx[x³/3] = x²).
  3. Apply the Fundamental Theorem: F(3) − F(0) = 27/3 − 0 = 9.
  4. Sanity check: the region sits inside the 3×9 bounding rectangle (area 27) and fills exactly one third — consistent with area 9. ✓

Answer: the area is 9 square units.

04 · Q&A

Test your understanding

Why is there always a "+ C" on an indefinite integral?

Because the derivative of any constant is 0, adding a constant to an antiderivative does not change its derivative. So if F(x) works, so does F(x) + 5 or F(x) − 100. The + C represents this whole family. In a definite integral the constant cancels in the subtraction, which is why it disappears there.

Evaluate ∫₁² (2x + 3) dx.

Antiderivative: x² + 3x. Evaluate: at 2, 4 + 6 = 10; at 1, 1 + 3 = 4. Difference 10 − 4 = 6. Geometrically this is the area of a trapezium under the line from x = 1 to x = 2.

What does it mean if a definite integral comes out negative?

It means the curve lies below the x-axis over that interval, so the "signed area" is negative. The integral counts area above the axis as positive and below as negative. To get the total physical area regardless of sign, integrate the absolute value, splitting the interval where the curve crosses the axis.

How are the derivative and the integral inverse to each other?

The Fundamental Theorem states it in two directions: differentiating an integral recovers the original function, d/dx ∫ₐˣ f(t) dt = f(x); and integrating a derivative recovers the original (up to endpoints), ∫ₐᵇ F′(x) dx = F(b) − F(a). One undoes the other, which is why every derivative rule has a matching integral rule.

Concept mind map

How the ideas connect

Every key idea in this chapter, branching from the core concept — use it to see the whole picture at a glance.

AntiderivativeIndefiniteintegralDefinite integralArea under curveFundamentaltheoremPower ruleintegralIntegral Calculus
Infographic

The process, step by step

Step 1Find the antiderivativeReverse differentiation using the power rule
Step 2Add C if indefiniteInclude the constant of integration
Step 3Evaluate at the boundsCompute F(b) - F(a) for a definite integral
Step 4Interpret the resultRead it as signed area under the curve
Solved examples

Worked problems, step by step

Follow each solution line by line, then try to reproduce it on paper before moving on.

Example 1Find the integral of 2x with respect to x.

  1. Raise the power and divide: x^2
  2. Add the constant of integration
  3. Result is x^2 + C

Example 2Find the area under y = x^2 from 0 to 3.

  1. Antiderivative is x^3/3
  2. Evaluate 3^3/3 - 0 = 27/3
  3. Area = 9
Practice problem set

Now you try

Work each one out first, then tap to reveal the worked answer.

1Integrate x^2.
x^3/3 + C.
2Integrate 6x + 1.
3x^2 + x + C.
3Evaluate the integral of 2 from 1 to 4.
2x from 1 to 4 gives 8 - 2 = 6.
4Find the area under y = 2x from 0 to 5.
x^2 from 0 to 5 gives 25.
5Integrate 1/x^2, that is x^(-2).
This gives -1/x + C.
6Evaluate the integral of x from 0 to 2.
x^2/2 from 0 to 2 gives 2.