Question

The area of the region under the curve of a function f(x)=axᵇon the interval [0, 4] is 16 square units. (a, b) ≠ __________.

a. (1, 2)
b. (2, 1)
c. (2, 2)
d. (1, 1)

Answer 1

The area under the curve of a function f(x) = ax^b on the interval [0, 4] can be found using integration. None of the given options satisfy the condition for the area to be 16 square units.

Step-by-step explanation:

The area under the curve of a function f(x) = ax^b on the interval [0, 4] can be found by evaluating the definite integral of the function over that interval. In this case, the area is given as 16 square units. To find the values of a and b that satisfy this condition, we can set up the definite integral and solve for a and b.

∫0⁴ (ax^b) dx = 16
∫0⁴ ax^b dx = 16
a/(b+1)(4^(b+1) - 0^(b+1)) = 16
a(4^(b+1)) = 16(b+1)
a = (16(b+1))/(4^(b+1))

So, a will have a value dependent on b. Therefore, none of the given options a. (1, 2), b. (2, 1), c. (2, 2), d. (1, 1) satisfy the condition for the area to be 16 square units.