To solve this problem, we will approach it in two parts:

Part 1: Estimating the area using $T_{10}$

We need to compute the average of the left- and right-endpoint Riemann sums using $N = 10$ rectangles.

The function is given by:

$f(x) = x + x^2$

The interval is from $x = 1$ to $x = 9$. The formula for the left-endpoint Riemann sum $L_N$ is:

$L_N = \sum_{i=0}^{N-1} f(a + i \cdot \Delta x) \cdot \Delta x$

And the formula for the right-endpoint Riemann sum $R_N$ is:

$R_N = \sum_{i=1}^{N} f(a + i \cdot \Delta x) \cdot \Delta x$

Where:

• $a = 1$
• $b = 9$
• $\Delta x = \frac{b - a}{N} = \frac{9 - 1}{10} = 0.8$

Now, the average of the left- and right-endpoint sums is:

$T_{10} = \frac{L_{10} + R_{10}}{2}$

We'll compute both $L_{10}$ and $R_{10}$, then find the average.

Part 2: Exact area using the Fundamental Theorem of Calculus

We need to find the exact area under the curve using the Fundamental Theorem of Calculus, which says:

$\int_a^b f(x) \, dx = F(b) - F(a)$

Where $F(x)$ is any antiderivative of $f(x)$.

Step 1: Find the antiderivative of $f(x) = x + x^2$.

$F(x) = \int (x + x^2) \, dx = \frac{x^2}{2} + \frac{x^3}{3}$

Step 2: Apply the Fundamental Theorem of Calculus:

$\int_1^9 (x + x^2) \, dx = F(9) - F(1)$

Now, we calculate $F(9)$ and $F(1)$.

$F(9) = \frac{9^2}{2} + \frac{9^3}{3} = \frac{81}{2} + \frac{729}{3}$ $F(1) = \frac{1^2}{2} + \frac{1^3}{3} = \frac{1}{2} + \frac{1}{3}$

Subtract these values to get the exact area.

Let me compute these values for you. Define the functions for calculating the Riemann sums and the integral def f(x): return x + x**2

Calculate left and right endpoint sums

a = 1 b = 9 N = 10 delta_x = (b - a) / N

Left-endpoint sum

L_10 = sum(f(a + i * delta_x) * delta_x for i in range(N))

Right-endpoint sum

R_10 = sum(f(a + (i + 1) * delta_x) * delta_x for i in range(N))

Average of the left- and right-endpoint sums

T_10 = (L_10 + R_10) / 2

Compute the exact integral using the Fundamental Theorem of Calculus

F = lambda x: (x2 / 2) + (x3 / 3)

Exact area under the curve

exact_area = F(b) - F(a)

T_10, exact_areaThe estimated area under the curve using the average of the left- and right-endpoint Riemann sums ($T_{10}$) is approximately:

$T_{10} = 283.520 \, \text{units}^2$

The exact area, calculated using the Fundamental Theorem of Calculus, is:

$\text{Exact Area} = 282.667 \, \text{units}^2$

Additional Questions:

1. How does the choice of $N$ (the number of rectangles) impact the accuracy of the Riemann sum estimate?
2. What is the difference between using left-endpoint and right-endpoint Riemann sums?
3. How can the trapezoidal rule provide a more accurate approximation than the left- and right-endpoint Riemann sums?
4. How would you calculate the exact area for a more complex function, say $f(x) = \sin(x)$?
5. Can you extend the use of the Fundamental Theorem of Calculus to functions with discontinuities?

Tip:

When calculating definite integrals, always check if the function has any discontinuities within the interval, as this can affect the applicability of the Fundamental Theorem of Calculus.