To find \(\int_{-1}^{1} f(x) \, dx\), we can use the given integrals and the properties of definite integrals. We know: \[ \int_{-4}^{5} f(x) \, dx = -8.0 \] \[ \int_{-4}^{-1} f(x) \, dx = -2.4 \] \[ \int_{1}^{5} f(x) \, dx = -6.1 \] We can break down the integral \(\int_{-4}^{5} f(x) \, dx\) into parts: \[ \int_{-4}^{5} f(x) \, dx = \int_{-4}^{-1} f(x) \, dx + \int_{-1}^{1} f(x) \, dx + \int_{1}^{5} f(x) \, dx \] Substitute the known values: \[ -8.0 = -2.4 + \int_{-1}^{1} f(x) \, dx + (-6.1) \] Combine the constants: \[ -8.0 = -2.4 - 6.1 + \int_{-1}^{1} f(x) \, dx \] \[ -8.0 = -8.5 + \int_{-1}^{1} f(x) \, dx \] Solve for \(\int_{-1}^{1} f(x) \, dx\): \[ \int_{-1}^{1} f(x) \, dx = -8.0 + 8.5 \] \[ \int_{-1}^{1} f(x) \, dx = 0.5 \] Therefore, the correct answer is: \[ \boxed{C} \]

To solve for \( \int_2^5 f(x) \, dx \) given that \( \int_0^5 f(x) \, dx = 3 \) and \( \int_0^2 f(x) \, dx = -2 \), follow these steps: 1. **Understand the relationship between the integrals**: The integral from 0 to 5 can be split into two parts: \[ \int_0^5 f(x) \, dx = \int_0^2 f(x) \, dx + \int_2^5 f(x) \, dx \] 2. **Substitute the given values**: \[ 3 = \int_0^2 f(x) \, dx + \int_2^5 f(x) \, dx \] Given that \( \int_0^2 f(x) \, dx = -2 \), substitute this value into the equation: \[ 3 = -2 + \int_2^5 f(x) \, dx \] 3. **Solve for \( \int_2^5 f(x) \, dx \)**: \[ 3 = -2 + \int_2^5 f(x) \, dx \] \[ \int_2^5 f(x) \, dx = 3 + 2 \] \[ \int_2^5 f(x) \, dx = 5 \] So, the value of \( \int_2^5 f(x) \, dx \) is \( 5 \). The correct answer is: - \( 5 \)

Let ff be a continuous function such that integral, from, 1, to, 6, of, f, of, x, d, x, equals, 2∫ 16​ f(x)dx=2 and integral, from, 6, to, minus, 6, of, f, of, x, d, x, equals, 8, .∫ 6−6​ f(x)dx=8. What is the value of integral, from, minus, 6, to, 1, of, f, of, x, d, x, question mark∫ −61​ f(x)dx? ​

Select the fourth function, y = 1x2 + 1, and set the interval to [−3, 2].(a) Find the approximate net area for 5 subintervals using left-endpoint rectangles. Find the approximate net area for 5 subintervals using right-endpoint rectangles.

Find the average value fave of the function f on the given interval.f(x) = 3x2 + 8x,    [−1, 5]

The graph of f is shown. Evaluate each integral by interpreting it in terms of areas.(a)    10f(x) dx0 (b)    25f(x) dx0 (c)    35f(x) dx25 (d)    45f(x) dx0