Published on: February 1, 2016 6:00 PM

Bookmark

**ENGG. MATHS - CALCULUS - CONCEPT CARD 3D**

**3D - INTEGRATION AND ITS APPLICATION**

**Integrals:**

The following are the two types of integrals:

**1. Definite Integrals:**

**Signed Area:**

A curve above the x-axis is said to enclose positive area, while a curve below the x-axis is said to enclose a negative area.

**Definite Integrals:**

The definite integral is the **signed area** between the function and the from x=a to x=b.

**2. Indefinite Integrals:**

**Anti Derivatives:**

The function F(x) is an antiderivative of f(x) if .

**Indefinite Integrals:**

The indefinite integral represents a family of antiderivatives:

**Fundamental Theorem of Calculus:**

**1. **If f(x) is continuous on the interval [a,b], then the area function is continuous and differentiable on the interval and .

**2. **If f(x) is continuous on the interval [a,b] and F(x) is any antiderivative of f(x), then .

**Approximations to the Area:**

**1. ****Left Hand Rectangle Approximation:**

**a. **The area under f(x) is approximated by a series of **n rectangles:**

**i.e. **the interval [a,b] is divided into **n equal sub-intervals** of **width** **to obtain (n+1) points** on the x-axis at:

**b. **Now the** 'n' points**,

**are the values of the function evaluated at the bottom left hand corners of the n rectancgles, **which are also taken as the **height of the corresponding rectangles**.

**c. **The **total area of n rectangles** then is:

**2. Right Hand Rectangle Approximation:**

**a. **The area under f(x) is approximated by a series of **n rectangles:**

**i.e. **the interval [a,b] is divided into **n equal sub-intervals** of **width** **to obtain (n+1) points** on the x-axis at:

**b. **Now the** 'n' points**,

**are the values of the function evaluated at the bottom right hand corners of the n rectancgles, **which are also taken as the **height of the corresponding rectangles**.

**c. **The **total area of n rectangles** then is:

**Note:**

**3. Midpoint Rule:**

The **height** of each rectangle can be taken to be the **value of the function evaluated at the midpoint of each rectangle**.

The **total area of n rectangles** is given by:

**4. Trapezoidal Rule:**

The area of a trapezoid is 1/2 X (sum of the parallel sides) X (distance between them).

The **total area of n trapezoids** is given by:

**5. Simpson's Rule:**

The **total area under f(x) **is given by:

Go to the previous Concept Card

**Dear Users, you are requested to bookmark Concept Cards on your Onlinetyari App so that you easily access all of them later under "My Favourite" tab*

Dear Aspirants, GATE 2016 paper was conducted successfully on 30th & 31st January 2016 for ME & EC disciplines. The question paper in the morning & forenoon session was of standard difficu ...

7 years agoIndian Government has decided to set up IARI in Assam The Indian Government has decided to set up an Indian Agricultural Research Institute in Kamrup district of Assam. Union Agriculture ministry has ...

7 years agoIndia joins IMF's Top Ten Elite Club India along with three other BRIC nations has joined the Bretton Woods-based International Monetary Fund’s (IMF) top-10 elite club. The IMF’s decision ...

7 years agoAll India Regional Editors Conference to begin in Jaipur In Rajasthan, two-day All India Regional Editors Conference has begun in Jaipur. Almost hundred editors from different parts of the country ar ...

7 years agoBanking Awareness Capsule 6: Reserve Bank of India Dear Aspirants, In the previous article, we have discussed the Scheduled commercial banks in detail and the number of commercial banks currently work ...

7 years agoK K Jalan appointed as MSME Secretary The Indian government has appointed Central Provident Fund Commissioner K K Jalan as Secretary, Ministry of Micro, Small and Medium Enterprises (MSME). He is a 1 ...

7 years ago© 2022 Indiavidual Learning Private Limited