Riemann Sum
Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus.
Riemann Integral
Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.
Find the area of the region bunded by the graphs of the given equations.
y= 2x+15 y=x^2
the area is
![### Finding the Area Between Curves
**Objective:**
Find the area of the region bounded by the graphs of the given equations.
**Equations:**
\( y = 2x + 15 \)
\( y = x^2 \)
**Area Calculation Steps:**
1. **Set the equations equal to find points of intersection:**
\[
x^2 = 2x + 15
\]
\[
x^2 - 2x - 15 = 0
\]
2. **Factor the quadratic equation:**
\[
(x + 3)(x - 5) = 0
\]
\[
x = -3, \quad x = 5
\]
3. **Define the functions for integration:**
\[
f(x) = 2x + 15
\]
\[
g(x) = x^2
\]
4. **Set up the integral for the area between the curves:**
\[
A = \int_{-3}^{5} [f(x) - g(x)] \, dx
\]
\[
A = \int_{-3}^{5} [2x + 15 - x^2] \, dx
\]
5. **Evaluate the integral:**
\[
\int [2x + 15 - x^2] \, dx = \left[ x^2 + 15x - \frac{1}{3}x^3 \right]_{-3}^{5}
\]
6. **Calculate the definite integral:**
- Plug in the limits \(x = 5\) and \(x = -3\) into the antiderivative to find the total area.
This calculation will provide the total area bounded between the two curves from \(x = -3\) to \(x = 5\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F267b304f-9594-4218-9c39-142b09d76524%2Fc71a78d2-e2b4-4d89-b9e2-5ad5327aa780%2F2snbkuj.jpeg&w=3840&q=75)
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