Questions: Find the area bounded by the graphs of the indicated equations over the given interval (when stated). Compute answers to three decimal places.
y=x^2+1; y=2x-2; -1 ≤ x ≤ 2
The area, calculated to three decimal places, is square units.
Transcript text: Find the area bounded by the graphs of the indicated equations over the given interval (when stated). Compute answers to three decimal places.
\[
y=x^{2}+1 ; y=2 x-2 ;-1 \leq x \leq 2
\]
The area, calculated to three decimal places, is $\square$ square units.
Solution
Solution Steps
Step 1: Define the Functions
We are given two functions:
f1(x)=x2+1f2(x)=2x−2
Step 2: Find the Points of Intersection
To find the area between the curves, we first need to determine the points where the two functions intersect. We solve the equation:
x2+1=2x−2
This simplifies to:
x2−2x+3=0
The solutions to this equation are:
x=1±2i
Since the solutions are complex, it indicates that the curves do not intersect in the real number domain over the interval [−1,2].
Step 3: Calculate the Area
Next, we calculate the area between the two curves over the interval [−1,2]. The area A is given by the integral:
A=∫−12∣f1(x)−f2(x)∣dx
Calculating this integral yields:
A=9
Final Answer
The area bounded by the graphs of the indicated equations over the interval [−1,2] is:
9 square units.