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Approximating Areas: Definite Integrals

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Wize University Calculus 1 Textbook > Integrals

The Definite Integral

4 Activities

Wize University Calculus 1 Textbook > Integrals

Approximating Areas & Riemann Sums

3 Activities

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?
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
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
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]
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 Approximating Areas & Riemann Sums Questions:
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]?
Practice: Area Using Sigma Notation (~F2014 Final Q17)
Practice Question: Area Using Sigma Notation
Determine the xx boundaries of the region whose area is lim⁡n→∞ Σi=1n 5n [8−(4+5in)2]\lim_{n\rightarrow\infty}\ \Sigma_{i=1}^n\ \frac{5}{n}\ \left[8-\left(4+\frac{5i}{n}\right)^2\right].
Practice: Area using Sigma Notation (~F2018 Final Q31)
Practice: Area using Sigma Notation
The area bounded by the function f(x)f\left(x\right) and the xx-axis in the interval [a, b]\left[a,\ b\right] is represented by lim⁡n→∞ Σi=1n(3n)[cos⁡(2+3in)(2+3in)]\lim_{n\rightarrow\infty}\ \Sigma_{i=1}^n\left(\frac{3}{n}\right)\left[\frac{\cos\left(2+\frac{3i}{n}\right)}{\left(2+\frac{3i}{n}\right)}\right] , find the function f(x)f\left(x\right) and the interval [a, b]\left[a,\ b\right].
Practice: Approximating Rectangles
Q.\textbf{Q.} Estimate the area under the curve y=1−x2y=1-x^2 on the interval [0,1][0,1] using 4 rectangles and left end-points.
Identify Integral from Definition: Riemann Sums
The given expression represents the limit of the sum of areas of nn rectangles approximating the area of a certain region in the plane. Identify f(x), Δxf\left(x\right),\ \Delta x and the interval [a,b]\left[a,b\right] which correspond to this expression, and write it as a definite integral: lim⁡n→∞∑i=1nsin⁡(πin)⋅πn\displaystyle \lim_{n\to\infty}\sum_{i=1}^n\sin\left(\frac{\pi i}{n}\right)\cdotp\frac{\pi}{n}
Approximating Rectangles: Riemann Sums
Approximate the area under the curve of the function f(x)=1xf(x)=\dfrac{1}{x} from x=2x=2 to x=5x=5 .
Approximating Rectangles: Riemann Sums
Approximate the area under the curve of the function f(x)=1xf(x)=\dfrac{1}{x} from x=2x=2 to x=5x=5 .
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 Σi=110 [2(i−3i2)−1]\Sigma_{i=1}^{10}\ \left[2\left(i-3i^2\right)-1\right].
Evaluate Σi=110 [2(i−3i2)−1]\Sigma_{i=1}^{10}\ \left[2\left(i-3i^2\right)-1\right].
Find the bounds of the region whose area is
lim⁡n→∞Σi=1n(4n)((−2+4in)−3(−2+4in)4)\lim_{n\rightarrow\infty}\Sigma_{i=1}^n\left(\frac{4}{n}\right)\left(\left(-2+\frac{4i}{n}\right)-3\left(-2+\frac{4i}{n}\right)^4\right)
Evaluate lim⁡x→∞Σi=1n 1n⋅[e(−3+in)]\lim_{x\rightarrow\infty}\Sigma_{i=1}^n\ \frac{1}{n}\cdot\left[e^{\left(-3+\frac{i}{n}\right)}\right].
Riemann Sums: Approximating Areas
Approximate the area under the curve of
f(x)=2−cos⁡2 (πx)f\left(x\right)=2-\cos^{2\ }\left(\pi x\right) on the interval [0,8]\left[0,8\right] using a left hand Riemann Sum and 4 sub intervals.
Approximate the area under the curve of
f(x)=2−cos⁡2 (πx)f\left(x\right)=2-\cos^{2\ }\left(\pi x\right) on the interval [0,8]\left[0,8\right] using a left hand Riemann Sum and 4 sub intervals.
Approximating Areas
Using 3 rectangles (partition of size 3) and right end points, the area under f(x)=9−xf(x)=\sqrt{9-x} on the interval [0, 9] is estimated to be
Approximating Area and Riemann Sums
Use 4 left rectangles in a Riemann sum to approximate the area under y=x2y=x^2 on [0,2][0,2].
Riemann Sums: Approximating Areas
The area bounded by the function f(x)f\left(x\right) and the xx-axis in the interval [a,b]\left[a,b\right] is given by lim⁡n→∞∑i=1n4in2[ein]\displaystyle \lim_{n\to\infty}\sum_{i=1}^n\frac{4i}{n^2}\left[e^{\frac{i}{n}}\right].
Determine f(x)f(x), aa, and bb.
Approximating area under curve
Practice: Area using Riemann Sums
The area bounded by the function f(x)f\left(x\right) and the xx-axis in the interval [a, b]\left[a,\ b\right] is represented by lim⁡n→∞ ∑i=1n(3n)[cos⁡(2+3in)(2+3in)]\displaystyle \lim_{n\rightarrow\infty}\ \sum_{i=1}^n\left(\frac{3}{n}\right)\left[\frac{\cos\left(2+\frac{3i}{n}\right)}{\left(2+\frac{3i}{n}\right)}\right].
Find the function f(x)f\left(x\right) and the interval [a, b]\left[a,\ b\right].
Practice: Estimating Area Using Riemann Sum
Given the following table of values for the function f(x)f\left(x\right), find the left-endpoint Riemann sum L3L_3 and right-endpoint Riemann sum R3R_3 to estimate the area under the graph of f(x)f\left(x\right) on the interval [0, 6]\left[0,\ 6\right].
Identify Integral from Definition: Riemann Sums
The given expression represents the limit of the sum of areas of nn rectangles approximating the area of a certain region in the plane. Identify f(x), Δxf\left(x\right),\ \Delta x and the interval [a,b]\left[a,b\right] which correspond to this expression, and write it as a definite integral: lim⁡n→∞∑i=1nsin⁡(πin)⋅πn\displaystyle \lim_{n\to\infty}\sum_{i=1}^n\sin\left(\frac{\pi i}{n}\right)\cdotp\frac{\pi}{n}
Approximating Rectangles: Riemann Sums
Approximate the area under the curve of the function f(x)=1xf(x)=\dfrac{1}{x} from x=2x=2 to x=5x=5 .
Practice: Approximating Rectangles
Q.\textbf{Q.} Estimate the area under the curve y=1−x2y=1-x^2 on the interval [0,1][0,1] using 4 rectangles and left end-points.
Evaluate lim⁡x→∞Σi=1n 1n⋅[e(−3+in)]\lim_{x\rightarrow\infty}\Sigma_{i=1}^n\ \frac{1}{n}\cdot\left[e^{\left(-3+\frac{i}{n}\right)}\right].
Evaluate Σi=110 [2(i−3i2)−1]\Sigma_{i=1}^{10}\ \left[2\left(i-3i^2\right)-1\right].
Find the bounds of the region whose area is
lim⁡n→∞Σi=1n(4n)((−2+4in)−3(−2+4in)4)\lim_{n\rightarrow\infty}\Sigma_{i=1}^n\left(\frac{4}{n}\right)\left(\left(-2+\frac{4i}{n}\right)-3\left(-2+\frac{4i}{n}\right)^4\right)
Riemann Sums: Approximating Areas
Use the left-endpoint Riemann sum, with n=4n=4, to estimate the area under the curve f(x)=ex−xf\left(x\right)=e^x-x on the interval [1, 13]\left[1,\ 13\right].
Riemann Sums: Approximating Areas
Which of the following integrals can be used to represent the limit lim⁡n→∞ Σi=1n[3n(3in−1)sin⁡(3in+2)]\lim_{n\to\infty}\ \Sigma_{i=1}^n\left[\frac{3}{n}\left(\frac{3i}{n}-1\right)\sin\left(\frac{3i}{n}+2\right)\right]?
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 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]?
Practice: Area Using Sigma Notation (~F2014 Final Q17)
Practice Question: Area Using Sigma Notation
Determine the xx boundaries of the region whose area is lim⁡n→∞ Σi=1n 5n [8−(4+5in)2]\lim_{n\rightarrow\infty}\ \Sigma_{i=1}^n\ \frac{5}{n}\ \left[8-\left(4+\frac{5i}{n}\right)^2\right].
Practice: Area using Sigma Notation (~F2018 Final Q31)
Practice: Area using Sigma Notation
The area bounded by the function f(x)f\left(x\right) and the xx-axis in the interval [a, b]\left[a,\ b\right] is represented by lim⁡n→∞ Σi=1n(3n)[cos⁡(2+3in)(2+3in)]\lim_{n\rightarrow\infty}\ \Sigma_{i=1}^n\left(\frac{3}{n}\right)\left[\frac{\cos\left(2+\frac{3i}{n}\right)}{\left(2+\frac{3i}{n}\right)}\right] , find the function f(x)f\left(x\right) and the interval [a, b]\left[a,\ b\right].
What is the integral that equals lim⁡n→∞Σi=1n(1n)[3(1+in)3+3]\lim_{n\rightarrow\infty}\Sigma_{i=1}^n\left(\frac{1}{n}\right)\left[3\left(1+\frac{i}{n}\right)^3+3\right]
What is the integral that equals lim⁡n→∞Σi=1n(1n)[3(1+in)3+3]\lim_{n\rightarrow\infty}\Sigma_{i=1}^n\left(\frac{1}{n}\right)\left[3\left(1+\frac{i}{n}\right)^3+3\right]