M171 Calculus I · Unit 4: Optimization & the integral · on Exam 4
§4.4 The Fundamental Theorem of Calculus
Study notes for The Fundamental Theorem of Calculus in M171 Calculus I at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- FTC (evaluation): if $F' = f$ on $[a,b]$, then $\int_a^b f(x)\,dx = F(b) - F(a)$. Any antiderivative works; the constant cancels.
- Total change theorem: $\int_a^b f'(x)\,dx = f(b) - f(a)$: integrating a rate gives the net change in the quantity.
- You need a table of basic antiderivatives (reverse the derivative rules). Check by differentiating.
Key formulas
FTC
$$\int_a^b f(x)\,dx = F(b) - F(a),\quad F' = f$$
Total change
$$\int_a^b f'(x)\,dx = f(b) - f(a)$$
Basic antiderivatives
$$\int x^n dx = \frac{x^{n+1}}{n+1} + C\ (n\ne-1),\ \int \frac1x dx = \ln|x| + C,\ \int e^x dx = e^x + C,\ \int \cos x\,dx = \sin x + C,\ \int\sin x\,dx = -\cos x + C$$
Worked example
Water enters a tank at $r(t) = 6t + 2$ gal/min. If the tank holds 10 gal at $t=0$, how much is in it at $t=4$?
Show the solution
Change $= \int_0^4 (6t+2)\,dt = \big[3t^2 + 2t\big]_0^4 = 48 + 8 = 56$ gal. Volume at $t=4$: $10 + 56 = 66$ gal.
Common mistakes
- Computing $F(a) - F(b)$.
- Antidifferentiating $\frac1x$ as $\frac{x^0}{0}$; it is $\ln|x|$.
Exam tipAlways differentiate your antiderivative in your head before evaluating. It takes five seconds and catches most errors.
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