To find the area under the curve defined by the equation [tex]\( y = x + 5x + 4 \)[/tex] over the interval [tex]\([1, 5]\)[/tex], follow these steps:
1. Simplify the given curve equation:
The given equation is:
[tex]\[
y = x + 5x + 4
\][/tex]
Combine like terms to simplify the equation:
[tex]\[
y = 6x + 4
\][/tex]
2. Set up the definite integral:
The area under the curve from [tex]\(x = 1\)[/tex] to [tex]\(x = 5\)[/tex] can be found by integrating the function [tex]\(y = 6x + 4\)[/tex] with respect to [tex]\(x\)[/tex] over the given interval [tex]\([1, 5]\)[/tex].
The integral we need to calculate is:
[tex]\[
\int_{1}^{5} (6x + 4) \, dx
\][/tex]
3. Integrate the function:
To find the integral, we need to find the antiderivative of [tex]\(6x + 4\)[/tex]. The antiderivative of [tex]\(6x\)[/tex] is [tex]\(3x^2\)[/tex], and the antiderivative of [tex]\(4\)[/tex] is [tex]\(4x\)[/tex]. Combining these, the antiderivative of [tex]\(6x + 4\)[/tex] is:
[tex]\[
\int (6x + 4) \, dx = 3x^2 + 4x + C
\][/tex]
4. Evaluate the definite integral:
We now need to evaluate the antiderivative at the upper and lower limits of the interval and subtract:
[tex]\[
\left[ 3x^2 + 4x \right]_{1}^{5}
\][/tex]
Calculate the value at the upper limit [tex]\( x = 5 \)[/tex]:
[tex]\[
3(5)^2 + 4(5) = 3(25) + 20 = 75 + 20 = 95
\][/tex]
Calculate the value at the lower limit [tex]\( x = 1 \)[/tex]:
[tex]\[
3(1)^2 + 4(1) = 3(1) + 4 = 3 + 4 = 7
\][/tex]
Subtract the value at the lower limit from the value at the upper limit:
[tex]\[
95 - 7 = 88
\][/tex]
Therefore, the area under the curve [tex]\(y = 6x + 4\)[/tex] over the interval [tex]\([1, 5]\)[/tex] is [tex]\(88\)[/tex] square units.
