Final answer:
To find the surface area of a cone using integration, we can imagine unrolling the curved surface of the cone and converting it into a flat shape. First, find the slant height of the cone using the Pythagorean theorem. Then, multiply the length of the arc by the slant height and divide by 2 to find the surface area.
Step-by-step explanation:
To find the surface area of a cone using integration, we can imagine unrolling the curved surface of the cone and converting it into a flat shape. This flat shape will be a sector of a circle with radius equal to the slant height of the cone and an arc length equal to the circumference of the base of the cone.
First, let's find the slant height of the cone using the Pythagorean theorem:
h² = r² + L²
L = sqrt(h² - r²)
The circumference of the base of the cone is 2πr. So, the length of the arc is 2πr.
Finally, we can find the surface area by multiplying the length of the arc by the slant height and dividing by 2:
Surface Area = (2πr * L) / 2