Answer: To find the area of the region between the graph of y = x^2 + 1 and the x-axis on the interval [2, 4], we can integrate the function f(x) = x^2 + 1 with respect to x over the given interval. The definite integral represents the area under the curve between the specified x-values. Here's how to calculate it using integration:
∫[2,4] (x^2 + 1) dx
To integrate this function, we apply the power rule for integration. The power rule states that the integral of x^n with respect to x is (x^(n+1))/(n+1), where n is any real number except -1.
∫(x^2 + 1) dx = [(x^3)/3 + x] + C
Now, we can evaluate the definite integral over the interval [2, 4]:
[(4^3)/3 + 4] - [(2^3)/3 + 2]
= (64/3 + 4) - (8/3 + 2)
= (64/3 + 12/3) - (8/3 + 6/3)
= (76/3) - (14/3)
= 62/3
Therefore, the area of the region between the graph of y = x^2 + 1 and the x-axis on the interval [2, 4] is 62/3 square units.
Explanation: