Question

Calculate the area between the graph of f(x) = x³ +4 and the x-axis given x elementof [0, 1].

a) 1
b) 11/3
c) 13/3
d) 17/4
e) 15/4

Answer 1

D) 17/4

Step-by-step explanation:

`\int\limits^0_1 {x^(3) + 4} \, dx = (1)/(4)x^(4) +4x\\((1)/(4) (0)^(4) + 4(0)) - ((1)/(4) (1)^(4)+4(1))\\(0)-((1)/(4) +4)\\(1)/(4) + (16)/(4) =(17)/(4)`

Answer 2

To calculate the area between the graph of f(x) = x³ + 4 and the x-axis given x elementof [0, 1], find the integral of f(x) over the interval. The area is found by evaluating the antiderivative at the upper and lower limits and subtracting the values. Option 1 is correct.

Step-by-step explanation:

To calculate the area between the graph of f(x) = x³ + 4 and the x-axis given x ∈ [0, 1], we need to find the integral of f(x) over the given interval. The area under the graph can be calculated by finding the definite integral of the function.

Let's calculate it step by step.

1. First, find the antiderivative of f(x) = x³ + 4. The antiderivative of x³ is (1/4)x⁴ and the antiderivative of 4 is 4x. So, the antiderivative of f(x) is (1/4)x⁴ + 4x.
2. Next, evaluate the antiderivative at the upper limit and lower limit of the given interval. At x = 1, the antiderivative is (1/4)(1)⁴ + 4(1) = 1 + 4 = 5. At x = 0, the antiderivative is (1/4)(0)⁴ + 4(0) = 0.
3. Finally, we have to subtract the value at the lower limit from the upper limit value. The area between the graph and the x-axis is 5 - 0 = 5.

Therefore, the correct answer is a) 1.