Answer :
The area under the function f(x) = 2[tex]x^{(1/2)[/tex] + 7 between 4 and 16 is 476/3 square units.
Here's how to find the area under the curve:
1. Set up the Definite Integral
The area under the curve f(x) = 2[tex]x^{(1/2)[/tex] + 7 between x = 4 and x = 16 is represented by the definite integral:
∫₄¹⁶ (2[tex]x^{(1/2)[/tex] + 7) dx
2. Evaluate the Integral
Find the antiderivative:
∫(2[tex]x^{(1/2)[/tex] + 7) dx = (4/3)[tex]x^{(3/2)[/tex] + 7x + C
Evaluate at the limits of integration:
[(4/3)[tex](16)^{(3/2)[/tex] + 7(16)] - [(4/3)[tex](4)^{(3/2)[/tex] + 7(4)]
Simplify:
[(4/3)(64) + 112] - [(4/3)(8) + 28]
= (256/3 + 112) - (32/3 + 28)
= (592/3) - (116/3)
= 476/3.
