M171 Calculus I · Unit 4: Optimization & the integral · on Exam 4
§4.3 The definite integral
Study notes for The definite integral 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 StudyCat.
Big ideas
- $\displaystyle\int_a^b f(x)\,dx = \lim_{n\to\infty}\sum_{i=1}^n f(x_i)\Delta x$: the net signed area between the curve and the $x$-axis on $[a,b]$.
- Properties: linearity, splitting at a point, reversing limits changes the sign, $\int_a^a f = 0$.
- Average value of $f$ on $[a,b]$ is $\frac{1}{b-a}\int_a^b f(x)\,dx$: the height of the rectangle with the same area.
- Use geometry (triangles, semicircles) to evaluate integrals of simple graphs exactly.
Key formulas
Properties
$$\int_a^b (cf+g) = c\!\int_a^b f + \int_a^b g,\quad \int_a^b f = \int_a^c f + \int_c^b f,\quad \int_b^a f = -\int_a^b f$$
Average value
$$f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx$$
Worked example
Given $\int_0^3 f = 4$ and $\int_3^5 f = -1$, find $\int_0^5 2f(x)\,dx$, $\int_5^0 f$, and the average value of $f$ on $[0,5]$.
Show the solution
$\int_0^5 2f = 2(4 + (-1)) = 6$. $\int_5^0 f = -3$. Average value $= \frac{1}{5}\cdot 3 = 0.6$.
Common mistakes
- Treating the definite integral as always positive: below the axis counts negative.
Exam tipDraw the region. Net signed area questions are far easier with a sketch.
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