FInd the area of the region bounded by the graphs of y= 3x^2+1 , y=0, x=-2.7, x=2.5
Make a sketch of the functions and you'll find that the area resembles two triangles whose total area is about 40.
Area =
∫(3x^2+1-0)dx for x from -2.7 to 2.5
I get 40.5 approx.
To find the area of the region bounded by the given graphs, we need to calculate the definite integral of the function y=3x^2+1 with respect to x, within the given limits.
First, let's plot the given equations on a graph to visualize the region.
The equation y = 3x^2 + 1 represents a parabolic curve that opens upward, and the equation y = 0 represents the x-axis. The two vertical lines x = -2.7 and x = 2.5 bound the region.
```
import matplotlib.pyplot as plt
import numpy as np
x = np.linspace(-3, 3, 100)
y = 3 * x**2 + 1
plt.plot(x, y, label='y = 3x^2 + 1')
plt.fill_between(x, y, where=(y >= 0), color='gray', alpha=0.3)
plt.fill_between(x, y, where=(y <= 0), color='white')
plt.axhline(0, color='black')
plt.axvline(-2.7, color='red', linestyle='--', label='x = -2.7')
plt.axvline(2.5, color='blue', linestyle='--', label='x = 2.5')
plt.xlabel('x')
plt.ylabel('y')
plt.xlim(-3, 3)
plt.ylim(-5, 20)
plt.legend()
plt.grid(True)
plt.show()
```
From the graph, we can see that the region we need to find the area of is the shaded region above the x-axis, bounded by the vertical lines x = -2.7 and x = 2.5.
To calculate the area, we integrate the function y = 3x^2 + 1 with respect to x between the x-limits -2.7 and 2.5:
```
Area = ∫[from -2.7 to 2.5] (3x^2 + 1) dx
```
The definite integral of 3x^2 with respect to x is x^3, and the integral of 1 with respect to x is x. Thus, we can calculate the area as follows:
```
Area = [x^3/3 + x] [from -2.7 to 2.5]
```
Evaluating the definite integral with the given x-limits, we get:
```
Area = [(2.5)^3/3 + 2.5] - [(-2.7)^3/3 + (-2.7)]
= (15.625/3 + 2.5) - (19.683/3 - 2.7)
≈ 5.208 + 6.561
≈ 11.769
```
Therefore, the area of the region bounded by the graphs of y = 3x^2 + 1, y = 0, x = -2.7, and x = 2.5 is approximately 11.769 square units.