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Notes
Practice Questions
\(\textbf{1)}\) Solve using trapezoidal approximation with n subintervals.
\(\,\,\,\,\,\,\,\,\,\,\displaystyle\int_{2}^{7}x^2 \,dx \,\,\, n=5 \)
\(\textbf{2)}\) Solve using trapezoidal approximation with n subintervals.
\(\,\,\,\,\,\,\,\,\,\,\displaystyle\int_{0}^{3}x^3 \,dx \,\,\, n=3 \)
\(\textbf{3)}\) Solve using trapezoidal approximation with n subintervals.
\(\,\,\,\,\,\,\,\,\,\,\displaystyle\int_{0}^{2\pi}\sin{x} \,dx \,\,\, n=4 \)
\(\textbf{4)}\) Solve using trapezoidal approximation with n subintervals.
\(\,\,\,\,\,\,\,\,\,\,\displaystyle\int_{0}^{2}5-x^2 \,dx \,\,\, n=4 \)
\(\textbf{5)}\) Solve using trapezoidal approximation with n subintervals.
\(\,\,\,\,\,\,\,\,\,\,\displaystyle\int_{1}^{5}\displaystyle \frac{1}{x+2} \,dx \,\,\, n=4 \)
\(\textbf{6)}\) Solve using trapezoidal approximation with n subintervals.
\(\,\,\,\,\,\,\,\,\,\,\displaystyle\int_{1}^{3}x \sin{x} \,dx \,\,\, n=4 \)
\(\textbf{7)}\) Solve using trapezoidal approximation with n subintervals.
\(\,\,\,\,\,\,\,\,\,\,\displaystyle\int_{0}^{6}\sqrt{x^2+4} \,dx \,\,\, n=4 \)
See Related Pages\(\)
\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Properties of Integrals}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{a}^{b}cf(x) \, dx=c\displaystyle \int_{a}^{b}f(x) \,dx…\)
\(\bullet\text{ Indefinite Integrals- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int x^n \, dx = \displaystyle \frac{x^{n+1}}{n+1}+C…\)
\(\bullet\text{ Indefinite Integrals- Trig Functions}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int \cos{x} \, dx=\sin{x}+C…\)
\(\bullet\text{ Definite Integrals}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{5}^{7} x^3 \, dx…\)
\(\bullet\text{ Integration by Substitution}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int (x^2+3)^3(2x) \,dx…\)
\(\bullet\text{ Area of Region Between Two Curves}\)
\(\,\,\,\,\,\,\,\,A=\displaystyle \int_{a}^{b}\left[f(x)-g(x)\right]\,dx…\)
\(\bullet\text{ Arc Length}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{a}^{b}\sqrt{1+\left[f'(x)\right]^2} \,dx…\)
\(\bullet\text{ Average Function Value}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{1}{b-a} \int_{a}^{b}f(x) \,dx\)
\(\bullet\text{ Volume by Cross Sections}\)
\(\,\,\,\,\,\,\,\,\text{Volume}=\displaystyle \int_{a}^{b}\left(\text{Area}\right) \, dx…\)
\(\bullet\text{ Disk Method}\)
\(\,\,\,\,\,\,\,\,V=\displaystyle \int_{a}^{b}\left[f(x)\right]^2\,dx…\)
\(\bullet\text{ Cylindrical Shells}\)
\(\,\,\,\,\,\,\,\,V=2 \pi \displaystyle \int_{a}^{b} y f(y) \, dy…\)
In Summary
The Trapezoidal Rule is a mathematical method used to approximate the definite integral of a function. It is based on the idea of dividing the region under the curve of the function into a series of trapezoids, and then summing up the areas of those trapezoids to estimate the total area under the curve.
We learn about the Trapezoidal Rule in Calculus because it is a useful tool for approximating definite integrals, which are important in many fields, such as physics, engineering, and economics. The Trapezoidal Rule is especially useful in cases where it is difficult or impossible to evaluate the definite integral of a function analytically.
The Trapezoidal Rule is typically taught in a Calculus course. It is usually introduced alongside other methods for evaluating definite integrals.
The Trapezoidal Rule is one of many methods that have been developed for approximating definite integrals. Other methods include left & right Riemann sums, midpoint rule, Simpson’s rule, and the Romberg method. Each of these methods has its own advantages and disadvantages, and the best method to use depends on the specific problem at hand.
Some related topics to the Trapezoidal Rule include definite integrals, approximation methods, numerical integration, and the fundamental theorem of calculus. Understanding these topics can help students gain a deeper understanding of the Trapezoidal Rule and how it fits into the larger context of Calculus.
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