Final answer:
To calculate the area between two curves, we need to find their intersection points, determine the integrand by subtracting the lower function from the upper function, and integrate with respect to x across the interval given by the intersection points. Integrating with respect to y is not directly feasible without further context or correction to the functions provided.
Step-by-step explanation:
Calculating the Area between Curves
To find the area of the region bounded by the functions f(x)=x²−5x+6 and g(x)=x²+32, and the x-axis when integrating with respect to y, we need to consider the following steps:
- Identify the intersection points of the two curves which will provide the limits of integration.
- Set the two functions equal to each other to solve for x-values where the curves intersect.
- Subtract the lower curve from the upper curve to find the integrand.
- Integrate this difference with respect to x over the interval determined by the intersection points.
Since we are asked to integrate with respect to y, we must express x in terms of y from both equations, if possible. However, due to a likely typo in the question since we can't integrate 'x²' with respect to 'y' directly, this step is not possible. Assuming a correction to the functions that makes them invertible to express x in terms of y, we would then follow through with the integration with respect to y within the common range of y-values for both functions.