How do you find the integral of a linear line?

Question

The equation given is y=1.1x with the domain: [0,13] and range: [0,12.9]

Answer 1

To solve this, we'll use the basic rule of definite integral ∫_a^bf(x)dx = |F(x)|_a^b= [F(b)−F(a)], where a=0, b=13, and f(x)dx = 1.1x(dx).

From there, we plan our integral over the interval 0 to 13.

I = ∫₀¹³1.1x(dx)

Then, we apply linearity to pull out the coefficient, 1.1) and use the power rule, ∫xⁿdx=(xⁿ⁺¹)/(n+1) with n=1, to calculate the integral of 1.1x as F(x) = 1.1|x²/2|.

1.1 ∫₀¹³x(dx) = 1.1|x²/2|₀¹³

Then we calculate [F(13) - F(0)].

1.1 [(13²/2)-(0²/2)] = 1.1 (84.5 - 0) = 92.95

Answer 2

the (indefinite) integral is 0.55x^2 + c

since f(0) = 0 per the given domain and range , then

0.55*0^2 + c = 0 which forces c=0, so there is no constant of integration.

the indefinite integral is F(x) = 0.55x^2

The area under the curve is then

F(13) - F(0) - 12.9 - 0 = 12.9