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Wize University Calculus 1 Textbook > Integrals

The Definite Integral

Approximating Areas & Riemann SumsPrevious Section
Properties of Definite IntegralsNext Section
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The Definite Integrals & Area Under a Curve

If we let the number of rectangles used in our Riemann Sum approximation grow to infinity, we can compute the exact area under a curve.
Area=lim⁡n→∞∑i=1nf(xi)Δx\displaystyle \boxed{\text{Area}=\lim_{n\to\infty}\sum_{i=1}^nf(x_i)\Delta x}


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Definite Integrals

The area under the graph of a function f(x)f(x) on an interval [a,b][a,b] is given by the definite integral.
∫abf(x)dx=lim⁡n→∞∑i=1nf(xi)Δx\boxed{\int_{a}^{b}f(x)\text{d}x=\lim_{n\rightarrow\infty}\sum_{i=1}^{n}f(x_i)\Delta x}

where Δx=b−an\Delta x=\frac{b-a}{n} and xi=a+iΔxx_i=a+i\Delta x


Wize Tip
The indefinite integral ∫f(x) dx\displaystyle \int f(x)\ dx is a function.
The definite integral ∫abf(x) dx\displaystyle \int_a^b f(x) \ dx is a number.

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Example: The Definition of the Definite Integral

Describe the area calculated by the following sum and compute the area.

lim⁡n→∞ ∑i=1n[1n(1+in)2]\displaystyle \lim_{n\rightarrow\infty}\ \sum_{i=1}^n\left[\frac{1}{n}\left(1+\frac{i}{n}\right)^2\right]

∫abf(x)dx=lim⁡n→∞∑i=1nf(xi)Δx\boxed{\int_{a}^{b}f(x)\text{d}x=\lim_{n\rightarrow\infty}\sum_{i=1}^{n}f(x_i)\Delta x}

From the limit, we see thatΔx=1n  \Delta x=\frac{1}{n}\ \ .

Since Δx=b−an\Delta x=\frac{b-a}{n}, we get that b−a=1b-a=1.

Recall that xi=a+iΔxx_i=a+i\Delta x

From the limit, we also see that f(xi)=(1+in)2f\left(x_i\right)=\left(1+\frac{i}{n}\right)^2.

So, xi=1+inx_i=1+\frac{i}{n}, meaning that a=1a=1.

b−a=1  ⟹  b=2b-a=1 \implies b=2

Therefore, this limit describes the area under the curve f(x)=x2f\left(x\right)=x^2 on the interval [1, 2]\left[1,\ 2\right].

lim⁡n→∞ ∑i=1n[1n(1+in)2]\displaystyle \lim_{n\rightarrow\infty}\ \sum_{i=1}^n\left[\frac{1}{n}\left(1+\frac{i}{n}\right)^2\right]

=lim⁡n→∞ ∑i=1n[1n(1+2in+i2n2)]=\displaystyle \lim_{n\rightarrow\infty}\ \sum_{i=1}^n\left[\frac{1}{n}\left(1+\frac{2i}{n}+\frac{i^2}{n^2}\right)\right]

=lim⁡n→∞ ∑i=1n(1n+2in2+i2n3)=\displaystyle \lim_{n\rightarrow\infty}\ \sum_{i=1}^n\left(\frac{1}{n}+\frac{2i}{n^2}+\frac{i^2}{n^3}\right)

=lim⁡n→∞(nn+2n(n+1)2n2+n(n+1)(2n+1)6n3)=\displaystyle \lim_{n\rightarrow\infty}\left(\frac{n}{n}+\frac{2n(n+1)}{2n^2}+\frac{n(n+1)(2n+1)}{6n^3}\right)

=lim⁡n→∞(1+n+1n+2n2+3n+16n2)=\displaystyle \lim_{n\rightarrow\infty}\left(1+\frac{n+1}{n}+\frac{2n^2+3n+1}{6n^2}\right)

=lim⁡n→∞(1+1+1n+13+12n+16n2)=\displaystyle \lim_{n\rightarrow\infty}\left(1+1+\frac{1}{n}+\frac{1}{3}+\frac{1}{2n}+\frac{1}{6n^2}\right)

=1+1+13=73\displaystyle=1+1+\frac{1}{3}=\boxed{\frac{7}{3}}


Thus ∫12x2dx=73\displaystyle \int_1^2x^2dx=\frac{7}{3}
If Δx=3n\Delta x=\frac{3}{n} andxi=iΔxx_i=i\Delta x, then what is the integral that represents

lim⁡n→∞∑i=1nxi(1+xi)2Δx\displaystyle \lim_{n\rightarrow\infty}\sum_{i=1}^nx_i\left(1+x_i\right)^2\Delta x


Practice: Definite Integrals Using Riemann Sums

Evaluate ∫04(2x2+3)dx\displaystyle \int_{0}^{4}(2x^2 +3)\text{d}x using Riemann sums.

Extra Practice
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.
🌶️ 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}}