Find area of the region under the curve y = 5 x^3 - 9 and above the x-axis, for 3¡Ü x ¡Ü 6. area = ??
To find the area of the region under the curve y = 5x^3 - 9 and above the x-axis for 3 ≤ x ≤ 6, you can use integration.
The area under a curve between two points can be found by integrating the equation of the curve within those limits.
First, let's find the integral of the given function y = 5x^3 - 9 with respect to x. The integral of a polynomial function can be found by increasing the exponent by 1 and dividing by the new exponent. So, the integral of 5x^3 is (5/4)x^4, and the integral of -9 is -9x.
Integrating the given function, we get:
∫(5x^3 - 9) dx = (5/4)x^4 - 9x
Now, to find the area between x = 3 and x = 6, we need to evaluate the integral between these limits.
Area = ∫[3, 6] (5/4)x^4 - 9x dx
Evaluating this integral between x = 3 and x = 6 will give us the area under the curve between these limits.
Area = [(5/4)(6)^4 - 9(6)] - [(5/4)(3)^4 - 9(3)]
After calculating this expression, you will obtain the area of the region under the curve between x = 3 and x = 6.