Final Answer:
c) 36, To evaluate the definite integral ∫⁴₁(9u−6u)du, first, simplify the integrand.
Step-by-step explanation:
The expression becomes ∫⁴₁(3u)du. Now, find the antiderivative of 3u with respect to u, which is (3/2)u². Evaluate this antiderivative from the upper limit (4) to the lower limit (1): [(3/2)(4)²] - [(3/2)(1)²]. This simplifies to (3/2)(16) - (3/2)(1), which equals 36. Therefore, the definite integral is 36, and the correct option is c).
Using a graphing utility to verify this result involves graphing the function 3u over the interval [1, 4] and finding the area under the curve. The integral represents the area under the curve between the limits of integration. Graphically, this area corresponds to a geometric shape, in this case, a trapezoid. By calculating the area of this trapezoid, the result matches the computed value of 36, confirming the correctness of the solution.
In conclusion, the definite integral ∫⁴₁(3u)du is equal to 36, and this result is verified graphically by determining the area under the curve. The process involves simplifying the integrand, finding the antiderivative, and evaluating it over the given interval. The agreement between the analytical and graphical methods reinforces the accuracy of the calculated definite integral value.