Calculate the area enclosed by this arch.

Question

An arch is to be constructed 16 meters high and 4 meters across at the base. The shape of the arch is given by the function f(x)=16-x^4

Calculate the area enclosed by this arch.

Answer 1

Area A = ∫_-2²(16 - x⁴)dx = 2(16x - x⁵/5)₀² = 2(32 - 32/5) = 256/5

Answer 2

16 meters high

4 meters base

parabolic arch is f(x) = 16 - x^4

integral of 16 -x^4

= 16x - x^5/5 evaluated from 0 to 2 = 16(2) -2^5/5 = 32 - 32/5 = 25.6

multiply by 2 to get = 51.2 square meters = area under the curve/arch

or evaluate from -2 to 2, the base = 4, from -2 to 2 = 4

16x - x^5/5

16(2) - (2)^5/5 - 16(-2) - (-2)^5/5

= 32 -32/5 +32 - 32/5

= 64-64/5

= 64- 12.8

= 51.2 square meters