Consider what happens if you modify your function to the form y=Ax^n (from y=X^n) what is the effect of changing the value of A on the area formula? Can you justify why this should be the case?
(from culculating curvy area)
To analyze the effect of changing the value of A on the area formula, let's understand the relationship between the original formula (y = X^n) and the modified formula (y = A * x^n).
In the original formula, y = X^n, we have a simple power relationship where the variable X is raised to the power of n. Here, X represents the independent variable, and y represents the dependent variable.
When we modify this function to y = A * x^n, we introduce a new constant, A, which multiplies the entire expression. A represents the coefficient or scale factor that affects the entire function.
To visualize the effect of changing the value of A on the area formula, let's assume we are finding the area under the curve created by y = A * x^n over a specific interval. The area would be obtained by integrating the function over that interval.
When A is increased, the entire curve is amplified vertically. If A is decreased, the entire curve is compressed vertically. The effect on the area will be proportional to the change in A.
Specifically, if A is multiplied by a factor of k, the area under the curve will also be multiplied by k. If A is divided by a factor of k, the area under the curve will be divided by k.
The justification for this relationship lies in the constant factor property of integration. While integrating an expression, if we multiply or divide the entire expression by a constant, the resulting integral will be multiplied or divided by the same constant.
So, modifying the value of A in the formula y = A * x^n scales the entire curve and proportionally scales the area under the curve.