a) Analytically:
To evaluate the definite integral ∫23x^3 dx, we can use the power rule of integration. According to the power rule, when integrating a term of the form x^n, where n is a real number except for -1, the result is (1/(n+1)) * x^(n+1).
Applying the power rule to our integral, we have:
∫23x^3 dx = [(1/4) * x^4] │2 to 3
= (1/4) * (3^4 - 2^4)
= (1/4) * (81 - 16)
= (1/4) * 65
= 65/4
= 16.25
Therefore, analytically, the value of the definite integral ∫23x^3 dx is 16.25.
b) Using five rectangles numerically:
To evaluate the integral numerically using five rectangles, we can use the midpoint rule of numerical integration. The midpoint rule states that the integral of a function can be approximated by evaluating the function at the midpoint of each subinterval and multiplying it by the width of the subinterval.
In this case, we want to divide the interval [2, 3] into five equal subintervals, each with a width of Δx = (3 - 2) / 5 = 1/5.
Using the midpoint rule, the approximate value of the integral is given by:
Δx * [f(x_1/2) + f(x_3/2) + f(x_5/2) + f(x_7/2) + f(x_9/2)]
where f(x) = x^3 and x_i/2 represents the midpoint of each subinterval.
Substituting the values into the formula, we get:
(1/5) * [f((2+3)/2) + f((2+2(1/5))/2) + f((2+3(1/5))/2) + f((2+4(1/5))/2) + f((2+5(1/5))/2)]
= (1/5) * [f(2.5) + f(2.3) + f(2.6) + f(2.8) + f(3.0)]
= (1/5) * [(2.5^3) + (2.3^3) + (2.6^3) + (2.8^3) + (3.0^3)]
Calculating this expression, we find:
= 0.2 * [15.625 + 12.167 + 17.576 + 22.912 + 27.000]
= 0.2 * 95.280
= 19.056
Therefore, using five rectangles numerically, the approximate value of the definite integral ∫23x^3 dx is 19.056.