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Notes
| \({\text{Properties of Integrals}}\) | |
| \(\underline{\text{Name}}\) | \(\underline{\text{Formula}}\) |
|---|---|
Questions & Solutions
\(\textbf{1)}\) \( \displaystyle \int_{2}^{2}f(x)\,dx \)
\(\textbf{2)}\) \( \displaystyle \int_{2}^{9}3\,dx \)
\(\textbf{3)}\) \( \displaystyle \int_{8}^{8}f(x)\,dx \)
\(\textbf{4)}\) \( \displaystyle \int_{1}^{10}5\,dx \)
\(\textbf{5)}\) \(\displaystyle \int_{2}^{5}f(x)\,dx=7, \enspace \displaystyle \int_{5}^{9}f(x)\,dx=4, \enspace \displaystyle \int_{2}^{12}f(x)dx=16 \)
\( \text{Solve for } \displaystyle \int_{2}^{9}f(x)\,dx \)
\(\textbf{6)}\) \(\displaystyle \int_{2}^{5}f(x)\,dx=7, \enspace \displaystyle \int_{5}^{9}f(x)\,dx=4, \enspace \displaystyle \int_{2}^{12}f(x)\,dx=16 \)
\( \text{Solve for } \displaystyle \int_{5}^{12}f(x)\,dx \)
\(\textbf{7)}\) \(\displaystyle \int_{2}^{5}f(x)\,dx=7, \enspace \displaystyle \int_{5}^{9}f(x)\,dx=4, \enspace \displaystyle \int_{2}^{12}f(x)\,dx=16 \)
\( \text{Solve for } \displaystyle \int_{9}^{5}f(x)\,dx \)
\(\textbf{8)}\) \(\displaystyle \int_{2}^{5}f(x)\,dx=7, \enspace \displaystyle \int_{5}^{9}f(x)\,dx=4, \enspace \displaystyle \int_{2}^{12}f(x)\,dx=16 \)
\( \text{Solve for } \displaystyle \int_{2}^{5}4f(x)\,dx \)
\(\textbf{9)}\) If\( \displaystyle \int_{a}^{b}f(x)\,dx=5a-2b \), then what is \( \displaystyle \int_{a}^{b}[f(x)-3]dx? \)
See Related Pages\(\)
\(\bullet\text{ Trapezoidal Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{b-a}{2n}\left[f(a)+2f(x_1)+2f(x_2)+…+2fx_{n-1}+f(b)\right]…\)
\(\bullet\text{ Properties of Integrals}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{a}^{b}cf(x) \, dx=c\displaystyle \int_{a}^{b}f(x) \,dx…\)
\(\bullet\text{ Indefinite Integrals- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int x^n \, dx = \displaystyle \frac{x^{n+1}}{n+1}+C…\)
\(\bullet\text{ Indefinite Integrals- Trig Functions}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int \cos{x} \, dx=\sin{x}+C…\)
\(\bullet\text{ Definite Integrals}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{5}^{7} x^3 \, dx…\)
\(\bullet\text{ Integration by Substitution}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int (x^2+3)^3(2x) \,dx…\)
\(\bullet\text{ Area of Region Between Two Curves}\)
\(\,\,\,\,\,\,\,\,A=\displaystyle \int_{a}^{b}\left[f(x)-g(x)\right]\,dx…\)
\(\bullet\text{ Arc Length}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{a}^{b}\sqrt{1+\left[f'(x)\right]^2} \,dx…\)
\(\bullet\text{ Average Function Value}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{1}{b-a} \int_{a}^{b}f(x) \,dx\)
\(\bullet\text{ Volume by Cross Sections}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Disk Method}\)
\(\,\,\,\,\,\,\,\,V=\displaystyle \int_{a}^{b}\left[f(x)\right]^2\,dx…\)
\(\bullet\text{ Cylindrical Shells}\)
\(\,\,\,\,\,\,\,\,V=2 \pi \displaystyle \int_{a}^{b} y f(y) \, dy…\)
\(\bullet\text{ Andymath Homepage}\)
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