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Sketch the function f defined by f(x)={

1,if x=1,
0, if x≠1,
​Then using the construction of Riemann integral show that ∫₀²
f(x)dx=0

User Moudiz
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1 Answer

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Final answer:

To show that ∫₀² f(x)dx=0 using the construction of Riemann integral, we need to partition the interval [0,2] into subintervals and evaluate the Riemann sums. Since f(x) is a piecewise function with a value of 1 at x=1 and 0 for all other x, the Riemann sum for each subinterval will be 0, except for the subinterval that contains x=1, where it will be 1. As we take the limit of the Riemann sums as the partition becomes finer, all the subintervals except the one containing x=1 will have a width of 0, resulting in the Riemann sum approaching 0. Therefore, ∫₀² f(x)dx=0.

Step-by-step explanation:

To show that ∫₀² f(x)dx=0 using the construction of Riemann integral, we need to partition the interval [0,2] into subintervals and evaluate the Riemann sums. Since f(x) is a piecewise function with a value of 1 at x=1 and 0 for all other x, the Riemann sum for each subinterval will be 0, except for the subinterval that contains x=1, where it will be 1. As we take the limit of the Riemann sums as the partition becomes finer, all the subintervals except the one containing x=1 will have a width of 0, resulting in the Riemann sum approaching 0. Therefore, ∫₀² f(x)dx=0.

User Dro
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