Final answer:
To calculate the area bound between the y-axis and the function y = x³ for -1 ≤ y ≤ 1 by integrating along the y-axis, we will integrate with respect to y and use the power rule of integration.
Step-by-step explanation:
To calculate the area bound between the y-axis and the function y = x³ for -1 ≤ y ≤ 1 by integrating along the y-axis, we will integrate with respect to y. Since y = x³, we can rewrite the equation as x = y^(1/3). The limits of integration are -1 and 1. So, the area is given by the integral of x with respect to y from -1 to 1:
A = ∫ x dy = ∫ (y^(1/3)) dy
Using the power rule of integration, we can find the antiderivative of y^(1/3), which is (3/4)y^(4/3). We can then evaluate the integral:
A = [ (3/4)y^(4/3) ] from -1 to 1
Substituting the limits of integration:
A = (3/4)(1^(4/3) - (-1)^(4/3)) = (3/4)(1 + 1) = (3/4)(2) = 3/2