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Definite Integral

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Wize University Calculus 1 Textbook > Integrals

The Definite Integral

4 Activities

Wize University Calculus 2 Textbook > Integration

Definite Integrals

5 Activities

Rewrite lim⁡n→∞∑i=1n12isin⁡(2in)\displaystyle \lim_{n\to\infty}\sum_{i=1}^n\frac{1}{2i}\sin\left(\frac{2i}{n}\right) as a definite integral
More The Definite Integral Questions:
Definite Integral: Integral from Definition
Evaluate lim⁡n→∞∑i=1n (3n)11+3in\displaystyle\lim_{n\rightarrow\infty}\sum_{i=1}^n\ \left(\frac{3}{n}\right)\frac{1}{\sqrt{1+\dfrac{3i}{n}}}
Practice: Definite Integral Using Riemann Sums
Evaluate ∫04(2x2+3)dx\displaystyle \int_{0}^{4}(2x^2 +3)\text{d}x using Riemann sums.
Definite Integral: Integral from Definition
Evaluate lim⁡n→∞∑i=1n (3n)11+3in\displaystyle\lim_{n\rightarrow\infty}\sum_{i=1}^n\ \left(\frac{3}{n}\right)\frac{1}{\sqrt{1+\dfrac{3i}{n}}}
Definite Integral: Integral from Definition
Evaluate lim⁡n→∞∑i=1n (3n)11+3in\displaystyle\lim_{n\rightarrow\infty}\sum_{i=1}^n\ \left(\frac{3}{n}\right)\frac{1}{\sqrt{1+\dfrac{3i}{n}}}
The Definite Integral: Riemann Sums
Use Riemann sums to evaluate the definite integral ∫24x(x+2) dx\displaystyle \int_{2}^{4}x(x+2)\,\text{d}x
Use the limit de finition of the integral to compute∫12x2 dx\displaystyle\int_1^2x^2\ dx
Riemann Sums with $x^3$: Definite Integrals
Use Riemann sums to evaluate the definite integral ∫13x2(x+3) dx\displaystyle \int_{1}^{3}x^2(x+3)\,\text{d}x .
Find ∫12(x+2)dx\int_1^2\left(x+2\right)dx using Riemann sum. [No credit will be given for any other method]
Approximating Areas: Definite Integrals
The limit lim⁡n→∞∑i=1n1n1−(−1+in)2\displaystyle\lim_{n\to\infty}\sum_{i=1}^n\frac{1}{n}\sqrt{1-\left(-1+\frac{i}{n}\right)^2} describes which of the following?
Riemann Sums
Evaluate lim⁡n→∞ ∑i=1n[5n[(1+5in)2−3]]\displaystyle \lim_{n\to\infty}\ \sum_{i=1}^n\left[\frac{5}{n}\left[\left(1+\frac{5i}{n}\right)^2-3\right]\right]
Evaluate ∫−10dx1−x2\int_{-1}^0\frac{dx}{\sqrt{1-x^2}}
A train travels at (t+1) m/s(t+1) \space m/s. How far does it travel after 4 seconds?
Definite Integral
Use Riemann sums to find the area under the curve f(x)=2x−2f(x)=2x-2 on the interval [1,4][1,4].
Riemann Sums: Approximation of areas under curves
If Δx=3n\Delta x=\frac{3}{n} andxi=iΔxx_i=i\Delta x, then what is the integral that represents lim⁡n→∞Σi=1nxi(1+xi)2Δx\lim_{n\rightarrow\infty}\Sigma_{i=1}^nx_i\left(1+x_i\right)^2\Delta x?
Practice: Definite Integral Using Riemann Sums
Q.\textbf{Q.} Evaluate ∫04(2x2+3)dx\displaystyle \int_{0}^{4}(2x^2 +3)\text{d}x using Riemann sums.
Express this summation as an integral: ∑i=1nπ4πcos⁡(11+πi2n) when n → ∞.\sum_{i=1}^n\frac{\pi}{4\pi}\cos\left(\frac{1}{1+\frac{\pi i}{2n}}\right)\ when\ n\ \rightarrow\ \infty.
Definite Integral
Rewrite lim⁡n→∞∑i=1n12isin⁡(2in)\displaystyle \lim_{n\to\infty}\sum_{i=1}^n\frac{1}{2i}\sin\left(\frac{2i}{n}\right) as a definite integral
Practice: Definite Integral and Riemann Sum
Rewrite lim⁡n→∞∑i=1n(3n)(11+3in)\displaystyle \lim_{n\rightarrow\infty}\sum_{i=1}^n\left(\frac{3}{n}\right)\left(\frac{1}{\sqrt{1+\frac{3i}{n}}}\right) as a definite integral.
🌶️ Evaluate the limit by recognizing the Riemann sum:
lim⁡n→∞∑i=1n27i3n4.\lim\limits_{n\rightarrow\infin}\displaystyle\sum_{i=1}^n\frac{27i^3}{n^4}.
Enter your answer as a fraction.
Riemann Sums: Definite Integrals
Find ∫12(x+2)dx\int_1^2\left(x+2\right)dx using Riemann sum. [No credit will be given for any other method]
Use Riemann sums to evaluate the definite integral ∫13x2(x+3) dx\displaystyle \int_{1}^{3}x^2(x+3)\,\text{d}x .
Definite Integrals
Compute
lim⁡n→∞3n∑i=1n11+3in\displaystyle \lim_{n\rightarrow \infin}\frac{3}{n}\sum_{i=1}^n\frac{1}{1+\frac{3i}{n}}
Use the limit de finition of the integral to compute∫12x2 dx\displaystyle\int_1^2x^2\ dx
final114
Express this summation as an integral: ∑i=1nπ4πcos⁡(11+πi2n) when n → ∞.\sum_{i=1}^n\frac{\pi}{4\pi}\cos\left(\frac{1}{1+\frac{\pi i}{2n}}\right)\ when\ n\ \rightarrow\ \infty. [Note that the answer is not unique]
final114
Find ∫12(x+2)dx\displaystyle\int_1^2\left(x+2\right)dx using Riemann sum. [No credit will be given for any other method]
Riemann Sums with $x^3$: Definite Integrals
Use Riemann sums to evaluate the definite integral ∫13x2(x+3) dx\displaystyle \int_{1}^{3}x^2(x+3)\,\text{d}x .
The Definite Integral: Riemann Sums
Use Riemann sums to evaluate the definite integral ∫24x(x+2) dx\displaystyle \int_{2}^{4}x(x+2)\,\text{d}x
Definite Integral: Integral from Definition
Evaluate lim⁡n→∞∑i=1n (3n)11+3in\displaystyle\lim_{n\rightarrow\infty}\sum_{i=1}^n\ \left(\frac{3}{n}\right)\frac{1}{\sqrt{1+\dfrac{3i}{n}}}
Evaluate ∫−10dx1−x2\int_{-1}^0\frac{dx}{\sqrt{1-x^2}}
More Definite Integrals Questions:
Definite Integral
Rewrite lim⁡n→∞∑i=1n12isin⁡(2in)\displaystyle \lim_{n\to\infty}\sum_{i=1}^n\frac{1}{2i}\sin\left(\frac{2i}{n}\right) as a definite integral
Properties of Definite Integrals
Suppose that f(−x)=f(x)f\left(-x\right)=f\left(x\right) and g(−x)=−g(x)g\left(-x\right)=-g\left(x\right). If ∫−102f(x)=5\int_{-1}^02f\left(x\right)=5, the value of ∫−11[3f(x)+x⋅sin⁡x⋅g(x)]dx=\int_{-1}^1\left[3f\left(x\right)+x\cdot\sin x\cdot g\left(x\right)\right]dx=
Properties of Definite Integrals
Given that both f(x)f\left(x\right) andf′(x)f'\left(x\right) are continuous everywhere, iff(1)=10f\left(1\right)=10 and ∫−31f′(x)dx=12\displaystyle \int_{-3}^1f'\left(x\right)dx=12, find f(−3)f\left(-3\right).
Practice: Definite Integrals and Area (~F2018 Final Q31)
Practice: Definite Integrals and Area
What is the integral that equals lim⁡n→∞Σi=1n(1n)[3(1+in)3+3]\displaystyle\lim_{n\rightarrow\infty}\Sigma_{i=1}^n\left(\frac{1}{n}\right)\left[3\left(1+\frac{i}{n}\right)^3+3\right]?