To solve this problem, let's go through the steps carefully.

Step 1: Analyze the Integral

The integral given is: $\int_0^2 (2x + 5) \, dx$ This integral represents the area under the curve $y = 2x + 5$ from $x = 0$ to $x = 2$.

Step 2: Identify the Equation of the Line

The function $y = 2x + 5$ is a linear equation with a slope of 2 and a y-intercept of 5. This line crosses the y-axis at $(0, 5)$ and continues upward with a slope of 2.

To find the corresponding point on the line at $x = 2$: $y = 2(2) + 5 = 4 + 5 = 9$ So, the line passes through $(0, 5)$ and $(2, 9)$.

Step 3: Determine the Correct Graph

The correct graph should display a shaded region between $x = 0$ and $x = 2$ under the line $y = 2x + 5$, from $y = 5$ to $y = 9$. Observing the provided graphs:

• The first option (far left) shows this exact region bounded between $x = 0$ and $x = 2$.
• The other options do not match the area from $x = 0$ to $x = 2$ under the curve $y = 2x + 5$.

Therefore, the first graph (leftmost option) is the correct one.

Step 4: Use a Geometric Formula to Evaluate the Integral

The region under $y = 2x + 5$ from $x = 0$ to $x = 2$ forms a trapezoid. The area of a trapezoid with parallel sides of lengths $a$ and $b$ and height $h$ is: $\text{Area} = \frac{1}{2} (a + b) \times h$ For our trapezoid:

• $a = 5$ (the y-value at $x = 0$),
• $b = 9$ (the y-value at $x = 2$),
• $h = 2$ (the distance between $x = 0$ and $x = 2$).

So, $\text{Area} = \frac{1}{2} (5 + 9) \times 2 = \frac{1}{2} \times 14 \times 2 = 14$

Thus, the value of the integral is $\int_0^2 (2x + 5) \, dx = 14$.

Would you like more details or have any questions?

Related Questions:

1. How would you compute the integral if the limits were from $x = 1$ to $x = 3$?
2. How do you find the area of a region under a curve that is not a straight line?
3. What other methods could we use to solve the integral if we didn't have a geometric shape?
4. How would the integral change if the slope of the line were different, say $y = 3x + 5$?
5. What is the significance of the slope and y-intercept in the function $y = 2x + 5$?

Tip:

When working with integrals of linear functions over a closed interval, visualize the graph. This often helps in recognizing shapes like triangles or trapezoids, making it easier to calculate areas geometrically.