Question

Limits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1 $${\int }_3^7(4x+6)dx$$

Short answer

Question: Evaluate the definite integral using right Riemann sums and Theorem 5.1: $${\int }_3^7(4x+6)dx$$ Answer: The definite integral is equal to 104.

Step-by-step solution

1. Theorem 5.1

Theorem 5.1 states that for any continuous function $f$ over the interval $[a,b]$, the definite integral of $f$ can be computed using Riemann sums: $${\int }_a^{b}f(x)dx=\underset{n\to \mathrm{\infty }}{lim}\sum _{i=1}^{n}f(x_i^{\ast })\mathrm{\Delta }x$$ where $x_i^{\ast }$ is an arbitrary point in the $i^{th}$ subinterval of the partition and $\mathrm{\Delta }x=(b-a)/n$. In our case, we are asked to use right Riemann sums, so $x_i^{\ast }$ will be the right endpoint of the $i^{th}$ subinterval.

2. Set up right Riemann sum

Our function is $f(x)=4x+6$, and we want to evaluate ${\int }_3^7(4x+6)dx$. Since we are using right Riemann sums, we will choose $x_i^{\ast }$ as the right endpoint of the $i^{th}$ subinterval. We will divide the interval $[3,7]$ into $n$ equal subintervals, and the width of each subinterval would be $\mathrm{\Delta }x=\frac{7-3}{n}=\frac{4}{n}$. Now let's write down the Riemann sum: $$R_n=\sum _{i=1}^{n}f(x_i^{\ast })\mathrm{\Delta }x$$ The right endpoint of the $i^{th}$ subinterval is $x_i^{\ast }=3+\frac{4i}{n}$. Since we have $f(x)=4x+6$, our Riemann sum becomes: $$R_n=\sum _{i=1}^n(4(3+\frac{4i}{n})+6)\frac{4}{n}$$

3. Compute the Riemann sum

Now let's compute the Riemann sum: $$R_n=\frac{4}{n}\sum _{i=1}^n(12+\frac{16i}{n}+6)$$ $$R_n=\frac{4}{n}\sum _{i=1}^n(18+\frac{16i}{n})$$ Now we separate the sum, and we have: $$R_n=4(\frac{18}{n}\sum _{i=1}^{n}1+\frac{16}{n^2}\sum _{i=1}^{n}i)$$

4. Evaluate the limit as $n\to \mathrm{\infty }$

To evaluate the definite integral, we need to find the limit of the Riemann sum as $n\to \mathrm{\infty }$: $${\int }_3^7(4x+6)dx=\underset{n\to \mathrm{\infty }}{lim}{R}_n$$ We first need to find the limits of the sums. We know that $\sum _{i=1}^{n}1=n$, and we have a formula for the sum of the first $n$ positive integers: $\sum _{i=1}^{n}i=\frac{n(n+1)}{2}$. Using these formulas, we get: $${\int }_3^7(4x+6)dx=\underset{n\to \mathrm{\infty }}{lim}4(\frac{18n}{n}+\frac{16n(n+1)}{2n^2})$$ Now we can simplify the expression and find the limit: $${\int }_3^7(4x+6)dx=\underset{n\to \mathrm{\infty }}{lim}4(18+\frac{8(n+1)}{n})$$ $${\int }_3^7(4x+6)dx=\underset{n\to \mathrm{\infty }}{lim}4(18+8(\frac{n}{n}+\frac{1}{n}))$$ $${\int }_3^7(4x+6)dx=4(18+8(1+0))$$ Finally, we get: $${\int }_3^7(4x+6)dx=4(18+8)=4(26)=104$$

Key concepts

Riemann sums

Riemann sums are a powerful tool in calculus used to approximate definite integrals. They work by breaking down the area under a curve into smaller rectangles, whose areas are easier to calculate. As the number of rectangles increases, the approximation becomes more accurate.

In practice, Riemann sums involve choosing specific sample points within each subinterval of a partitioned domain to represent the height of the rectangles. This approach lets us estimate the total area under the curve by summing the areas of all these rectangles. Here's how it connects to definite integrals:

• The interval $[a,b]$ is divided into $n$ subintervals.
• The width of each subinterval is represented by $\mathrm{\Delta }x$.
• For each subinterval, we calculate the height of the rectangle using a sample point $x_i^{\ast }$.

Riemann sums are foundational in understanding how integration works because they form the basis for the formal limit definition of definite integrals, connecting concrete geometry with abstract calculus concepts.

right Riemann sum

A right Riemann sum is a specific type of Riemann sum where the height of each rectangle is determined by the value of the function at the right endpoint of each subinterval. This choice influences the approximation since different endpoints can yield different sums.

In a right Riemann sum, given a function $f(x)$ over an interval $[a,b]$:

• \