# Integrate with respect to x

• cabellos
In summary, integration with respect to x involves finding the area under a curve in the x-direction by adding up infinitely small rectangles. It differs from integration with respect to y in the direction of measurement. Its purpose is to solve problems involving rates of change, and it is the inverse operation of differentiation. Real-life applications of integration with respect to x include physics, engineering, economics, and statistics for finding areas, volumes, and averages and solving problems involving rates of change.
cabellos
I just need a check on when integrating sin(xy) with respect to x? Does this become -cos(xy) or -cos(x)

thanks

Neither. y is to be treated as a constant. And from that follows?

ok....is it -cos(xy) / y

yes that's correct.

## 1. What does it mean to integrate with respect to x?

Integrating with respect to x means finding the area under a curve in the x-direction. It involves adding up all the infinitely small rectangles that make up the curve.

## 2. How is integration with respect to x different from integration with respect to y?

The main difference is the direction in which the area is being measured. Integrating with respect to x measures the area in the x-direction, while integrating with respect to y measures the area in the y-direction.

## 3. What is the purpose of integrating with respect to x?

Integrating with respect to x is used to solve problems involving rates of change, such as finding the distance traveled by an object with varying velocity or the volume of a three-dimensional shape with varying cross-sectional area.

## 4. How is integration with respect to x related to derivatives?

Integration with respect to x is the inverse operation of differentiation, which is finding the rate of change of a function. Integration allows us to recover the original function from its derivative by finding the area under the curve of the derivative.

## 5. Are there any applications of integration with respect to x in real life?

Integrating with respect to x has many real-life applications, such as in physics, engineering, economics, and statistics. It is used to solve problems involving rates of change and to find areas, volumes, and averages in various contexts.

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