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🌶️ Evaluate the limit by recognizing the Riemann sum: lim_n⭢∑_i=1^n(27i^3)/(n^…

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Wize University Calculus 1 Textbook > Integrals

The Fundamental Theorem of Calculus - Part 2

6 Activities

Wize University Calculus 1 Textbook > Integrals

The Definite Integral

4 Activities

🌶️ 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.
More The Fundamental Theorem of Calculus - Part 2 Questions:
Find the definite integral ∫02x(x+1)dx\displaystyle\int_0^2x(x+1)dx.
Practice: Property of Definite Integral
Practice: Property of Definite Integral
Given that bothf(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).
Fundamental Theorem of Calculus
Given that both f(x)f(x) and f′(x)f'(x) are continuous everywhere, and given f(1)=10f(1)=10 and ∫−31f′(x)dx=12\displaystyle\int_{-3}^1f'\left(x\right)dx=12 , find f(−3)f(-3).
Definite Integrals with Trig
Compute: ∫25(x3+sin⁡x)dx\displaystyle \int_{2}^{5}(x^3+\sin{x})\text{d}x
Integrals
Evaluate ∫−30∣(x−1)(x+2)∣dx\int_{-3}^0\left|\left(x-1\right)\left(x+2\right)\right|dx
Trigonometric Functions
If f(x)=arcsin⁡(ln⁡x)f\left(x\right)=\arcsin\left(\ln\sqrt{x}\right), then ∫1ef′(x)dx=\int_1^ef'\left(x\right)dx=
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]
Find ∫0π∣cos⁡ x∣dx\displaystyle\int_0^{\pi}\left|\cos\ x\right|dx
Practice: Property of Definite Integral
Given that bothf(x)f\left(x\right) andf′(x)f'\left(x\right) are continuous everywhere, iff(1)=10f\left(1\right)=10and ∫−31f′(x)dx=12\int_{-3}^1f'\left(x\right)dx=12, find f(−3)f\left(-3\right).
A factory produces pollution at a rate of(t+1) gm/hr(t+1)\ gm/hr. How much does it produce after 4 hours?
Find the definite integral ∫02x(x+1)dx\displaystyle\int_0^2x(x+1)dx.
Integrals
Evaluate ∫13x3+xxdx\int_1^3\frac{\sqrt{x^3}+x}{x}dx
If ∫02[3(kx)2+(k+1)x+1]dx=10\int_0^2[3(kx)^2+(k+1)x+1]dx=10 then what is a possible value of kk ?
Practice: Property of Definite Integral
Given that bothf(x)f\left(x\right) andf′(x)f'\left(x\right) are continuous everywhere, iff(1)=10f\left(1\right)=10and ∫−31f′(x)dx=12\int_{-3}^1f'\left(x\right)dx=12, find f(−3)f\left(-3\right).
Given that f(7)=1  and  f(5)=8f\left(7\right)=1\ \ and\ \ f\left(5\right)=8
find ∫57f′(x) dx\displaystyle \int_{5}^{7} f'(x)\space dx
Derivatives
Evaluate
∫0π(3x2+sin⁡x) dx\int_0^{\pi} (3x^2 + \sin x ) \ dx
Derivatives
Find
∫01(x4−2x3) dx\int_{0}^1(x^4-2x^3)\ dx
Antiderivatives: Trigonometric Functions
Find the definite integral ∫π/6π/3(sec⁡x+tan⁡x)sec⁡xdx\int_{\pi/6 }^{\pi/3}(\sec x+\tan x)\sec x dx
Practice: Definite Integral
Evaluate ∫−20∣x2−1∣dx\displaystyle \int_{-2}^0\left|x^2-1\right|dx
Practice: Definite Integral
Given that bothf(x)f\left(x\right) andf′(x)f'\left(x\right) are continuous everywhere, iff(1)=10f\left(1\right)=10and ∫−31f′(x)dx=12\displaystyle \int_{-3}^1f'\left(x\right)dx=12, find f(−3)f\left(-3\right).
Practice: Definite Integral
Evaluate ∫0319+x2+2xln⁡2 dx\displaystyle \int_0^3\frac{1}{9+x^2}+2^x\ln2\ dx.
Evaluate the integral using the Fundamental Theorem of Calculus: ∫01e2x+e−2xdx\displaystyle\int_0^1e^{2x}+e^{-2x}dx.
Evaluate the integral using the Fundamental Theorem of Calculus ∫242−416−x2dx\displaystyle\int_2^{\frac{4}{\sqrt{2}}}-\frac{4}{\sqrt{16-x^2}}dx.
final114
Find the definite integral ∫π/6π/3(sec⁡x+tan⁡x)sec⁡xdx\int_{\pi/6 }^{\pi/3}(\sec x+\tan x)\sec x dx
Find the exact value of
∫0πsin⁡xdx\int_0^{\pi}\sin xdx
Definite Integrals with Trig
Compute: ∫25(x3+sin⁡x)dx\displaystyle \int_{2}^{5}(x^3+\sin{x})\text{d}x
Fundamental Theorem of Calculus
Given that both f(x)f(x) and f′(x)f'(x) are continuous everywhere, and given f(1)=10f(1)=10 and ∫−31f′(x)dx=12\displaystyle\int_{-3}^1f'\left(x\right)dx=12 , find f(−3)f(-3).
final114
Find the definite integral ∫π/6π/3(sec⁡x+tan⁡x)sec⁡xdx\int_{\pi/6 }^{\pi/3}(\sec x+\tan x)\sec x dx
Evaluate the integral using the Fundamental Theorem of Calculus ∫242−416−x2dx\displaystyle\int_2^{\frac{4}{\sqrt{2}}}-\frac{4}{\sqrt{16-x^2}}dx.
Find ∫0π∣cos⁡ x∣dx\displaystyle\int_0^{\pi}\left|\cos\ x\right|dx
Practice: Property of Definite Integral
Practice: Property of Definite Integral
Given that bothf(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: Property of Definite Integral
Practice: Property of Definite Integral
Given that bothf(x)f\left(x\right) andf′(x)f'\left(x\right) are continuous everywhere, iff(1)=10f\left(1\right)=10and ∫−31f′(x)dx=12\int_{-3}^1f'\left(x\right)dx=12, find f(−3)f\left(-3\right).
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]
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.
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}}