Short on time? This page is the recap. For explanations, the diagrams and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: ∫ (a to x) f(t) dt means the definite integral of f(t) from t = a to t = x.
Recap
- A definite integral with a variable upper limit defines a new function: g(x) = ∫ (a to x) f(t) dt, the signed area under f from a to x.
- t is a dummy variable. x is the input of g and must not appear inside the integral.
- g(a) = 0. If x < a, the area is collected from right to left and its sign is reversed.
- Theorem: if f is continuous on an interval containing a, then g′(x) = f(x) on that interval.
- Reason: g(x + h) − g(x) is a thin strip of area about f(x) · h. Divide by h and let h → 0.
- Consequence: every continuous function has an antiderivative, even if no formula can be written for it.
Key relationships
| Expression | Derivative with respect to x |
|---|---|
| ∫ (a to x) f(t) dt | f(x) |
| ∫ (x to b) f(t) dt | −f(x) |
| ∫ (a to u(x)) f(t) dt | f(u(x)) · u′(x) |
| ∫ (u(x) to b) f(t) dt | −f(u(x)) · u′(x) |
| ∫ (c to x) f(t) dt, any other constant c | f(x) as well: changing the constant start only adds a constant |
Values of g come from signed areas (geometry). Slopes of g come from heights of f.
Assumptions behind the method
- f is continuous on an interval that contains the constant limit and every value of x (or u(x)) used.
- Corners in the graph of f are allowed; jumps are not.
- The chain rule form needs u to be differentiable.
Mistakes to avoid
- Differentiating the integrand instead of substituting x into it.
- Writing g(a) = f(a). It is 0.
- Missing the factor u′(x) when the upper limit is a function of x.
- Missing the minus sign when x is the lower limit.
- Adding unsigned areas when part of f is below the axis.
- Writing x inside the integral as well as in a limit.
Quick self-check
- d/dx ∫ (1 to x) (t³ + 2t) dt = ? (x³ + 2x)
- d/dx ∫ (x to 4) sin(√t) dt = ? (−sin(√x), for x > 0)
- d/dx ∫ (0 to 5x) cos(t²) dt = ? (5 cos(25x²))
- If g(x) = ∫ (3 to x) f(t) dt, what is g(3)? (0)
Next: practice questions.