Assume that a>0. Evaluate the integral from 0 to a of x dx by approximating the region bounded by y=x and the x-axis from 0 to a with rectangles. Use equal subintervals and take right endpoints.
To evaluate the integral from 0 to a of x dx by approximating the region with rectangles and using right endpoints, we need to apply the Right Endpoint Riemann Sum method.
The first step is to determine the number of subintervals. Since we are using equal subintervals, we can choose any number of subintervals. Let's denote the number of subintervals as n.
The width of each subinterval, Δx, can be calculated by dividing the total interval (a - 0 = a) by the number of subintervals:
Δx = a / n
Next, we find the x-coordinate of the right endpoint of each subinterval. This can be done by starting at the left endpoint (0) and adding Δx successively for each subinterval:
x_i = 0 + i * Δx, where i ranges from 1 to n.
Now, we approximate the area of each rectangle by multiplying the width Δx by the height, which is the value of x at the right endpoint. Therefore, the area of each rectangle is Δx * x_i.
To evaluate the integral, we sum up the areas of all rectangles by taking the limit as the number of subintervals approaches infinity:
∫₍₀₎ˣ x dx ≈ lim (n→∞) Σ Δx * x_i
The summation can be written as:
∫₍₀₎ˣ x dx ≈ lim (n→∞) Σ (a/n) * (i * (a/n))
Simplifying the expression:
∫₍₀₎ˣ x dx ≈ lim (n→∞) (a/n) * (Σ i * (a/n))
We can rewrite the summation using the formula for the sum of the first n natural numbers:
∫₍₀₎ˣ x dx ≈ lim (n→∞) (a/n) * (n * (n + 1) / 2) * (a/n)
Simplifying further:
∫₍₀₎ˣ x dx ≈ lim (n→∞) (a²/n²) * (n² + n) / 2
Taking the limit:
∫₍₀₎ˣ x dx ≈ lim (n→∞) (a² + a)/ (2n)
Finally, we substitute n with ∞, and we get the result:
∫₍₀₎ˣ x dx = (a² + a) / 2
Therefore, the integral from 0 to a of x dx, using the Right Endpoint Riemann Sum method and approximating with rectangles, is equal to (a² + a) / 2.