To find the area under the curve of the function y = 9x³ from x = 1 to x = t, we need to calculate the definite integral of the function over that interval. The formula for the area A under the curve from a to b is given by:
A = ∫ab f(x) dx
In this case, our function f(x) is 9x³, so we need to evaluate:
A = ∫1t 9x³ dx
Now, we find the antiderivative of 9x³. The antiderivative can be computed as follows:
∫ 9x³ dx = 9 * (1/4)x⁴ = (9/4)x⁴ + C
Where C is the constant of integration. Since we are working with a definite integral, we can ignore the constant C for our calculations. Next, we evaluate the definite integral:
A = [(9/4)x⁴] from 1 to t = (9/4)(t⁴) – (9/4)(1⁴)
Substituting the limits of integration:
A = (9/4)t⁴ – (9/4)(1) = (9/4)t⁴ – (9/4)
Therefore, the area under the curve y = 9x³ from x = 1 to x = t is:
A = (9/4)(t⁴ – 1)
In conclusion, just plug in the value of t to compute the area under the curve for any specific interval.