If `int_(0)^(3)f(x)dx=1` and `int_(0)^(4)f(y)dy=2`, then `int_(3)^(4)f(z)dz=`

Options

A. 7
B. `-1`
C. 1
D. 3

Answer

To solve the problem step by step, we will use the properties of definite integrals.

### Step-by-Step Solution:

1. **Given Integrals**:
We have the following information:
\[ \int_{0}^{3} f(x) \, dx = 1 \]
\[ \int_{0}^{4} f(y) \, dy = 2 \]

2. **Understanding the Integral from 3 to 4**:
We want to find the value of:
\[ \int_{3}^{4} f(z) \, dz \]
We can use the property of definite integrals which states:
\[ \int_{a}^{b} f(t) \, dt = \int_{a}^{c} f(t) \, dt + \int_{c}^{b} f(t) \, dt \]
where \( c \) is a point between \( a \) and \( b \).

3. **Applying the Property**:
Here, we can express the integral from 0 to 4 as:
\[ \int_{0}^{4} f(y) \, dy = \int_{0}^{3} f(y) \, dy + \int_{3}^{4} f(y) \, dy \]
Substituting the known values:
\[ 2 = 1 + \int_{3}^{4} f(y) \, dy \]

4. **Solving for the Unknown Integral**:
Rearranging the equation gives us:
\[ \int_{3}^{4} f(y) \, dy = 2 - 1 = 1 \]

5. **Final Result**:
Therefore, the value of the integral \( \int_{3}^{4} f(z) \, dz \) is:
\[ \int_{3}^{4} f(z) \, dz = 1 \]

### Summary:
The value of \( \int_{3}^{4} f(z) \, dz \) is \( 1 \).