Let (a, b) and `(lambda,mu)` be two points on the curve y = f(x) . If the slope of the tangent to the curve at (x,y) be `phi(x)`, then `int_(a)^lambda phi(x) dx` is

Options

A. `lambda-a`
B. `mu-b`
C. `lambda+mu-a-b`
D. none of these

Answer

To solve the problem, we need to evaluate the definite integral \(\int_{a}^{\lambda} \phi(x) \, dx\) given that \(\phi(x)\) is the slope of the tangent to the curve \(y = f(x)\) at the point \((x, y)\).

### Step-by-step Solution:

1. **Understanding the Points on the Curve**:
- We know that the points \((a, b)\) and \((\lambda, \mu)\) lie on the curve \(y = f(x)\).
- Therefore, we can write:
\[ b = f(a) \quad \text{and} \quad \mu = f(\lambda) \]

2. **Identifying the Slope**:
- The slope of the tangent to the curve at any point \(x\) is given by \(\phi(x)\).
- By definition, the slope of the curve \(y = f(x)\) can be expressed as:
\[ \phi(x) = \frac{dy}{dx} = f'(x) \]

3. **Setting Up the Integral**:
- We need to evaluate the integral:
\[ \int_{a}^{\lambda} \phi(x) \, dx = \int_{a}^{\lambda} f'(x) \, dx \]

4. **Applying the Fundamental Theorem of Calculus**:
- According to the Fundamental Theorem of Calculus, the integral of a derivative over an interval gives us the difference of the function values at the endpoints of the interval:
\[ \int_{a}^{\lambda} f'(x) \, dx = f(\lambda) - f(a) \]

5. **Substituting the Values**:
- Now substituting the values we found earlier:
\[ f(\lambda) = \mu \quad \text{and} \quad f(a) = b \]
- Therefore, we have:
\[ \int_{a}^{\lambda} \phi(x) \, dx = \mu - b \]

### Final Answer:
\[ \int_{a}^{\lambda} \phi(x) \, dx = \mu - b \]